A 61-cm-diameter wheel accelerates uniformly about its

Express the following angles in radians: (a) 45.0, (b) 60.0, (c) 90.0, (d) 360.0, and (e) 445. Give as numerical values and as fractions of

The Sun subtends an angle of about 0.5 to us on Earth, 150 million km away. Estimate the radius of the Sun.

A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle (Fig. 840) of What diameter spot will it make on the Moon?

The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 4.0 s. What is the angular acceleration as the blades slow down?

The platter of the hard drive of a computer rotates at per minute (a) What is the angular velocity of the platter? (b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? (c) If a single bit requires of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

A child rolls a ball on a level floor 3.5 m to another child. If the ball makes 12.0 revolutions, what is its diameter?

A grinding wheel 0.35 m in diameter rotates at 2200 rpm. Calculate its angular velocity in rad s. (b) What are the linear speed and acceleration of a point on the edge of the grinding wheel?

A bicycle with tires 68 cm in diameter travels 9.2 km. How many revolutions do the wheels make?

Calculate the angular velocity (a) of a clocks second hand, (b) its minute hand, and (c) its hour hand. State in rad s. (d) What is the angular acceleration in each case?

A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 841). (a) What is the linear speed of a child seated 1.2 m from the center? (b) What is her acceleration (give components)?

What is the linear speed, due to the Earths rotation, of a point (a) on the equator, (b) on the Arctic Circle (latitude 66.5 N), and (c) at a latitude of 42.0 N?

Calculate the angular velocity of the Earth (a) in its orbit around the Sun, and (b) about its axi

How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 gs?

A 61-cm-diameter wheel accelerates uniformly about its center from 120 rpm to 280 rpm in 4.0 s. Determine (a) its angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating.

In traveling to the Moon, astronauts aboard the Apollo spacecraft put the spacecraft into a slow rotation to distribute the Suns energy evenly (so one side would not become too hot). At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. Think of the spacecraft as a cylinder with a diameter of 8.5 m rotating about its cylindrical axis. Determine (a) the angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 6.0 min after it started this acceleration.

A turntable of radius is turned by a circular rubber roller of radius in contact with it at their outer edges. What is the ratio of their angular velocities

An automobile engine slows down from 3500 rpm to 1200 rpm in 2.5 s. Calculate (a) its angular acceleration, assumed constant, and (b) the total number of revolutions the engine makes in this time

A centrifuge accelerates uniformly from rest to 15,000 rpm in 240 s. Through how many revolutions did it turn in this time?

Pilots can be tested for the stresses of flying high-speed jets in a whirling human centrifuge, which takes 1.0 min to turn through 23 complete revolutions before reaching its final speed. (a) What was its angular acceleration (assumed constant), and (b) what was its final angular speed in rpm?

A cooling fan is turned off when it is running at It turns 1250 revolutions before it comes to a stop. (a) What was the fans angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop?

A wheel 31 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.8 s. How far will a point on the edge of the wheel have traveled in this time?

The tires of a car make 75 revolutions as the car reduces its speed uniformly from to The tires have a diameter of 0.80 m. (a) What was the angular acceleration of the tires? If the car continues to decelerate at this rate, (b) how much more time is required for it to stop, and (c) how far does it go?

A small rubber wheel is used to drive a large pottery wheel. The two wheels are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of and it is in contact with the pottery wheel (radius 27.0 cm) without slipping. Calculate (a) the angular acceleration of the pottery wheel, and (b) the time it takes the pottery wheel to reach its required speed of 65 rpm.

A 52-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. (a) What is the maximum torque she exerts? (b) How could she exert more torque?

Calculate the net torque about the axle of the wheel shown in Fig. 842. Assume that a friction torque of 0.60 mN opposes the motion.

A person exerts a horizontal force of 42 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted (a) perpendicular to the door and (b) at a 60.0 angle to the face of the door?

Two blocks, each of mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 843. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system when it is first released

The bolts on the cylinder head of an engine require tightening to a torque of If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm across (Fig. 844), estimate the force applied near each of the six points by a wrench.

Determine the net torque on the 2.0-m-long uniform beam shown in Fig. 845. All forces are shown. Calculate about (a) point C, the CM, and (b) point P at one end. 26. (II) A person exerts a horizontal force of 42 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted (a) perpendicular to the door and (b) at a 60.0 angle to the face of the door? 27. (II) Two blocks, each of mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 843. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system when it is first released. 18 N 35 N 28 N 24 cm 12 cm 135 FIGURE 8;42 Problem 25. m m l1 l2 FIGURE 8;43 Problem 27. 28. (II) The bolts on the cylinder head of an engine require tightening to a torque of If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm across (Fig. 844), estimate the force applied near each of the six points by a wrench. 95 mN. 28 cm 15 mm on bolt on wrench F B F B FIGURE 8;44 Problem 28. 65

(I) Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center.

Estimate the moment of inertia of a bicycle wheel 67 cm in diameter. The rim and tire have a combined mass of 1.1 kg. The mass of the hub (at the center) can be ignored (why?).

A merry-go-round accelerates from rest to in 34 s. Assuming the merry-go-round is a uniform disk of radius 7.0 m and mass 31,000 kg, calculate the net torque required to accelerate it.

(II) An oxygen molecule consists of two oxygen atoms whose total mass is \(5.3 \times 10^{-26}\ kg\) and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 \times \(10^{-46}\ kg \cdot m^2\). From these data, estimate the effective distance between the atoms.

(II) A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.380 kg. Calculate (a) its moment of inertia about its center, and (b) the applied torque needed to accelerate it from rest to 1750 rpm in 5.00 s. Take into account a frictional torque that has been measured to slow down the wheel from 1500 rpm to rest in 55.0 s.

The forearm in Fig. 846 accelerates a 3.6-kg ball at by means of the triceps muscle, as shown. Calculate (a) the torque needed, and (b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm

Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 846. The ball is accelerated uniformly from rest to in 0.38 s, at which point it is released. Calculate (a) the angular acceleration of the arm, and (b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.7 kg and rotates like a uniform rod about an axis at its end

A softball player swings a bat, accelerating it from rest to in a time of 0.20 s. Approximate the bat as a 0.90-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it

A small 350-gram ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculate (a) the moment of inertia of the ball about the center of the circle, and (b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore air resistance on the rod and its moment of inertia.

Calculate the moment of inertia of the array of point objects shown in Fig. 847 about (a) the y axis, and (b) the x axis. Assume and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the x axis. (c) About which axis would it be harder to accelerate this array?

(II) A potter is shaping a bowl on a potter’s wheel rotating at constant angular velocity of 1.6 rev/s (Fig.8-48). The friction force between her hands and the clay is 1.5 N total. (a) How large is her torque on the wheel, if the diameter of the bowl is 9.0 cm? (b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hands? The moment of inertia of the wheel and the bowl is \(0.11 kg \cdot m^2\).

A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 560 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?

A 0.72-m-diameter solid sphere can be rotated about an axis through its center by a torque of which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere?

(II) Let us treat a helicopter rotor blade as a long thin rod, as shown in Fig. 8–49. (a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. (b) How much torque must the motor apply to bring the blades from rest up to a speed of 6.0 rev/s in 8.0 s?

A centrifuge rotor rotating at 9200 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of If the mass of the rotor is 3.10 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest, and how long will it take?

To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 850. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 32 rpm in 5.0 min, starting from rest?

Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia I. The blocks move (towards the right) with an acceleration of along their frictionless inclines (see Fig. 851). (a) Draw free-body diagrams for each of the two blocks and the pulley. (b) Determine and the tensions in the two parts of the string. (c) Find the net torque acting on the pulley, and determine its moment of inertia, I.

(III) An Atwood machine consists of two masses, \(m_A=65 \ \mathrm {kg}\) and \(m_B=75 \ \mathrm {kg}\), connected by a massless inelastic cord that passes over a pulley free to rotate, Fig. 8–52. The pulley is a solid cylinder of radius R = 0.45 m and mass 6.0 kg. (a) Determine the acceleration of each mass. (b) What % error would be made if the moment of inertia of the pulley is ignored? [Hint: The tensions \(F_\mathrm{TA}\) and \(F_\mathrm{TB}\) are not equal. We discussed the Atwood machine in Example 4–13, assuming I = 0 for the pulley.]

(III) A hammer thrower accelerates the hammer (mass = 7.30 kg) from rest within four full turns (revolutions) and releases it at a speed of 26.5 m/s. Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate (a) the angular acceleration, (b) the (linear) tangential acceleration, (c) the centripetal acceleration just before release, (d) the net force being exerted on the hammer by the athlete just before release, and (e) the angle of this force with respect to the radius of the circular motion. Ignore gravity.

(I) An automobile engine develops a torque of \(265 \ \mathrm m \cdot \mathrm N\) at 3350 rpm. What is the horsepower of the engine?

A centrifuge rotor has a moment of inertia of How much energy is required to bring it from rest to 8750 rpm?

Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.20 m high. Assume it starts from rest and rolls without slipping

(II) A bowling ball of mass 7.25 kg and radius 10.8 cm rolls without slipping down a lane at 3.10 m/s. Calculate its total kinetic energy.

Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (a) that due to its daily rotation about its axis, and (b) that due to its yearly revolution about the Sun. [Assume the Earth is a uniform sphere with and is

A rotating uniform cylindrical platform of mass 220 kg and radius 5.5 m slows down from to rest in 16 s when the driving motor is disconnected. Estimate the power output of the motor (hp) required to maintain a steady speed of 3.8 rev/s?

A merry-go-round has a mass of 1440 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 7.00 s? Assume it is a solid cylinder

A sphere of radius and mass starts from rest and rolls without slipping down a 30.0 incline that is 10.0 m long. (a) Calculate its translational and rotational speeds when it reaches the bottom. (b) What is the ratio of translational to rotational kinetic energy at the bottom? Avoid putting in numbers until the end so you can answer: (c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

(II) A ball of radius r rolls on the inside of a track of radius R (see Fig. 8–53). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping?

Two masses, and are connected by a rope that hangs over a pulley (as in Fig. 854). The pulley is a uniform cylinder of radius and mass 3.1 kg. Initially is on the ground and rests 2.5 m above the ground. If the system is released, use conservation of energy to determine the speed of just before it strikes the ground. Assume the pulley bearing is frictionless.

A 1.80-m-long pole is balanced vertically with its tip on the ground. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.

What is the angular momentum of a 0.270-kg ball revolving on the end of a thin string in a circle of radius 1.35 m at an angular speed of 10.4 rad /s?

(a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 28 cm when rotating at 1300 rpm? (b) How much torque is required to stop it in 6.0 s?

A person stands, hands at his side, on a platform that is rotating at a rate of If he raises his arms to a horizontal position, Fig. 855, the speed of rotation decreases to (a) Why? (b) By what factor has his moment of inertia changed?

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed Assuming no external torques, what is the final common angular speed of the two disks?

A diver (such as the one shown in Fig. 828) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed (rev s) when in the straight position?

A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of If her initial moment of inertia was what is her final moment of inertia? How does she physically accomplish this change?

(a) What is the angular momentum of a figure skater spinning at with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 48 kg? (b) How much torque is required to slow her to a stop in 4.0 s, assuming she does not move her arms?

A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia The platform rotates without friction with angular velocity The person walks radially to the edge of the platform. (a) Calculate the angular velocity when the person reaches the edge. (b) Calculate the rotational kinetic energy of the system of platform plus person before and after the persons walk.

(II) A potter’s wheel is rotating around a vertical axis through its center at a frequency of 1.5 rev/s. The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 2.6-kg chunk of clay, approximately shaped as a flat disk of radius 7.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it? Ignore friction.

A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of Its total moment of inertia is Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. (a) What is the angular velocity of the merry-go-round now? (b) What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

(II) A uniform horizontal rod of mass M and length rotates with angular velocity \(\omega\) about a vertical axis through its center. Attached to each end of the rod is a small mass m. Determine the angular momentum of the system about the axis.

Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, (a) what would the Suns new rotation rate be? Take the Suns current period to be about 30 days. (b) What would be its final kinetic energy in terms of its initial kinetic energy of today?

(II) A uniform disk turns at 3.3 rev/s around a frictionless central axis. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–56. They then turn together around the axis with their centers superposed. What is the angular frequency in rev/s of the combination?

An asteroid of mass traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, in the direction of Earths rotation, and is embedded there. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.

Suppose a 65-kg person stands at the edge of a 5.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of The turntable is at rest initially, but when the person begins running at a speed of (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.

A merry-go-round with a moment of inertia equal to and a radius of 2.5 m rotates with negligible friction at A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation, causing the platform to slow to What is her mass?

A 1.6-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.

On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius and winds its way out to radius To read the digital information, a CD player rotates the CD so that the players readout laser scans along the spirals sequence of bits at a constant linear speed of Thus the player must accurately adjust the rotational frequency f of the CD as the laser moves outward. Determine the values for f (in units of rpm) when the laser is located at and when it is at R2 R .

(a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.013 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest. (b) What fraction of its kinetic energy is rotational?

A cyclist accelerates from rest at a rate of How fast will a point at the top of the rim of the tire be moving after 2.25 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at restsee Fig. 857.]

Suppose David puts a 0.60-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle, accelerating it from rest to a rate of 75 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?

Bicycle gears: (a) How is the angular velocity of the rear wheel of a bicycle related to the angular velocity of the front sprocket and pedals? Let and be the number of teeth on the front and rear sprockets, respectively, Fig. 858. The teeth are spaced the same on both sprockets and the rear sprocket is firmly attached to the rear wheel. (b) Evaluate the ratio when the front and rear sprockets have 52 and 13 teeth, respectively, and (c) when they have 42 and 28 teeth.

Figure 859 illustrates an molecule. The bond length is 0.096 nm and the bonds make an angle of 104. Calculate the moment of inertia of the molecule (assume the atoms are points) about an axis passing through the center of the oxygen atom (a) perpendicular to the plane of the molecule, and (b) in the plane of the molecule, bisecting the bonds.

A hollow cylinder (hoop) is rolling on a horizontal surface at speed when it reaches a 15 incline. (a) How far up the incline will it go? (b) How long will it be on the incline before it arrives back at the bottom?

Determine the angular momentum of the Earth (a) about its rotation axis (assume the Earth is a uniform sphere), and (b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun)

A wheel of mass M has radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 860). The step has height h, where What minimum force F is needed?

If the coefficient of static friction between a cars tires and the pavement is 0.65, calculate the minimum torque that must be applied to the 66-cm-diameter tire of a 1080-kg automobile in order to lay rubber (make the wheels spin, slipping as the car accelerates). Assume each wheel supports an equal share of the weight

A 4.00-kg mass and a 3.00-kg mass are attached to opposite ends of a very light 42.0-cm-long horizontal rod (Fig. 861). The system is rotating at angular speed about a vertical axle at the center of the rod. Determine (a) the kinetic energy KE of the system, and (b) the net force on each mass

A small mass m attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table (Fig. 8–62). Initially, the mass revolves with a speed \(v_1 = 2.4\ m/s\) in a circle of radius \(r_1 = 0.80\ m\). The string is then pulled slowly through the hole so that the radius is reduced to \(r_2 = 0.48\ m\). What is the speed, \(v_2\), of the mass now?

A uniform rod of mass M and length can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 863. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 820g.]

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrownoff mass carries off either (a) no angular momentum, or (b) its proportional share of the initial angular momentum.

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance , holding onto it, Fig. 864. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spools center of mass move?

The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moons spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

A spherical asteroid with radius r = 123 m and mass \(M = 2.25 \times 10^{10}\ kg\) rotates about an axis at four revolutions per day. A “tug” spaceship attaches itself to the asteroid’s south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the asteroid’s surface as shown in Fig. 8–65. If F = 285 N, how long will it take the tug to rotate the asteroid’s axis of rotation through an angle of \(5.0^{\circ}\) by this method?

Most of our Solar System’s mass is contained in the Sun, and the planets possess almost all of the Solar System’s angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar System’s total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass \(1.99 \times 10^{30}\ kg\), radius \(6.96 \times 10^8\ m\)) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planet’s spin about its own axis.

Water drives a waterwheel (or turbine) of radius R = 3.0 m as shown in Fig. 8-66. The water enters at a speed \(v_1 = 7.0\ m/s\) and exits from the waterwheel at a speed \(v_2 = 3.8\ m/s\). (a) If 85 kg of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5 s, how much power is delivered to the wheel?

The radius of the roll of paper shown in Fig. 867 is 7.6 cm and its moment of inertia is A force of 3.5 N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque of is exerted on the roll which gradually brings it to a stop. Assuming that the papers thickness is negligible, calculate (a) the length of paper that unrolls during the time that the force is applied (1.3 s) and (b) the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving

Problem 1COQ A solid ball and a solid cylinder roll down a ramp. They both start from rest at the same time and place. Which gets to the bottom first? (a) They get there at the same time. (b) They get there at almost exactly the same time except for frictional differences. (c) The ball gets there first. (d) The cylinder gets there first. (e) Can’t tell without knowing the mass and radius of each.

Bonnie sits on the outer rim of a merry-go-round, and Jill sits midway between the center and the rim. The merry go-round makes one complete revolution every 2 seconds. Jill’s linear velocity is: (a) the same as Bonnie’s. (b) twice Bonnie’s. (c) half of Bonnie’s. (d) one-quarter of Bonnie’s. (e) four times Bonnie’s.

Problem 1P (I) Express the following angles in radians: (a) 45.0°, (b) 60.0°, (c) 90.0°, (d) 360.0°, and (e) 445°. Give as numerical values and as fractions of ?

A bicycle odometer (which counts revolutions and is calibrated to report distance traveled) is attached near the wheel axle and is calibrated for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Why are Eqs. 8–4 and 8–5 valid for radians but not for revolutions or degrees? Read Section 8–1 and follow the derivations carefully to find the answer.

An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle in time t, through what angle did it rotate in the time \(\frac{1}{2}t\)? (a) \(\frac{1}{2} \theta.\) (b) \(\frac{1}{4} \theta.\) (c) \(\theta.\) (d) \(2 \theta.\) (e) \(4 \theta.\)

Problem 2P (I) The Sun subtends an angle of about 0.5° to us on Earth, 150 million km away. Estimate the radius of the Sun.

Suppose a disk rotates at constant angular velocity. (a) Does a point on the rim have radial and or tangential acceleration? (b) If the disk’s angular velocity increases uniformly, does the point have radial and or tangential acceleration? (c) For which cases would the magnitude of either component of linear acceleration change?

Problem 2SL Total solar eclipses can happen on Earth because of amazing coincidences: for one, the sometimes near-perfect alignment of Earth, Moon, and Sun. Secondly, using the information inside the front cover, calculate the angular diameters (in radians) of the Sun and the Moon, as seen from Earth, and then .

A car speedometer that is supposed to read the linear speed of the car uses a device that actually measures the angular speed of the tires. If larger-diameter tires are mounted on the car instead, how will that affect the speedometer reading? The speedometer (a) will still read the speed accurately. (b) will read low. (c) will read high.

(I) A laser beam is directed at the Moon, from Earth. The beam diverges at an angle \(\theta\) (Fig. 8-40) of \(1.4 \times 10^{-5}\ \mathrm{rad}\). What diameter spot will it make on the Moon? ________________ Equation Transcription: Text Transcription: theta 1.4x10^{-5} rad theta

Can a small force ever exert a greater torque than a larger force? Explain.

Problem 3SL Two uniform spheres simultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. (a) Which reaches the bottom of the incline first? (b) Which has the greater speed there? (c) Which has the greater total kinetic energy at the bottom? Explain your answers

The solid dot shown in Fig. is a pivot point. The board can rotate about the pivot. Which force shown exerts the largest magnitude torque on the board? FIGURE 8-36 MisConceptual Question 4 .

Problem 4P (I) The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 4.0 s. What is the angular acceleration as the blades slow down?

Why is it more difficult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this.

A bicyclist traveling with speed \(v=8.2 \ \mathrm{m} / \mathrm{s}\) on a flat road is making a turn with a radius \(r=13 \ \mathrm{m}\). There are three forces acting on the cyclist and cycle: the normal force \(\left(\vec{F}_{N}\right)\) and friction force \(\left(\vec{F}_{f r}\right)\) exerted by the road on the tires; and \(m \vec{g}\), the total weight of the cyclist and cycle. Ignore the small mass of the wheels. (a) Explain carefully why the angle \(\theta\) the bicycle makes with the vertical (Fig. 8-68) must be given by \(\tan \theta=F_{f r} / F_{N}\) if the cyclist is to maintain balance. (b) Calculate \(\theta\) for the values given. [Hint: Consider the "circular" translational motion of the bicycle and rider.] (c) If the coefficient of static friction between tires and road is \(\mu_{s}\) = 0.65, what is the minimum turning radius? Equation Transcription: v=8.2 m/s r=13 m () () tan = Text Transcription: v=8.2 m/s r=13 m (vector F_N) (vector F_fr) m vector g theta tan theta = F_fr/F_N mu_s

Consider a force \(F=80 \ \mathrm{N}\) applied to a beam as shown in Fig. 8-37. The length of the beam is \(l=5.0 \ \mathrm{m}\), \(\text { anc } \theta=37^{\circ}\), so that \(x=3.0 \ \mathrm{m}\) and \(y=4.0 \ \mathrm{m}\). Of the following expressions, which ones give the correct torque produced by the force \(\vec{F}\) around point \(P\)? (a) \(80 N\) (b) \((80 \ \mathrm{N})(5.0 \ \mathrm{m})\) (c) \((80 \ N)(5.0 \ m)\left(\sin 37^{\circ}\right)\) (d) \((80 \ N)(4.0 \ \mathrm{m})\) (e) \((80 \ \mathrm{N})(3.0 \ \mathrm{m})\) (f) \((48 \ N)(5.0 \ m)\) (g) \((48 \ N)(4.0 \ \mathrm{m})\left(\sin 37^{\circ}\right)\) Equation Transcription: F=80 N l=5.0 m anc =37° x=3.0 m y=4.0 m P 80 N (80 N)(5.0 m) (80 N)(5.0 m)(sin?37°) (80 N)(4.0 m) (80 N)(3.0 m) (48 N)(5.0 m) (48 N)(4.0 m)(sin?37°) Text Transcription: F=80 N l=5.0 m anc theta =37^circ x=3.0 m y=4.0 m vector F P 80 N (80 N)(5.0 m) (80 N)(5.0 m)(sin?37^circ) (80 N)(4.0 m) (80 N)(3.0 m) (48 N)(5.0 m) (48 N)(4.0 m)(sin?37^circ)

Problem 5P (II) The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions) per minute = rev/min) (a) What is the angular velocity (rad/s )of the platter? (b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? (c) If a single bit requires 0.50 µ m of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

If the net force on a system is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero? Explain and give examples.

Problem 5SL One possibility for a low-pollution automobile is for it to use energy stored in a heavy rotatingflywheel. Suppose such a car has a total mass of 1100 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 270 kg, and should be able to travel 350 km without needing a flywheel “spinup.” (a) Make reasonable assumptions (average frictional retarding force on car = 450 N, thirty acceleration periods from rest 95 km/h, to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and estimate what total energy needs to be stored in the flywheel. (b) What is the angular velocity of the flywheel when it has a full “energy charge”? (c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip?

Two spheres have the same radius and equal mass. One sphere is solid, and the other is hollow and made of a denser material.Which one has the bigger moment of inertia about an axis through its center? (a) The solid one. (b) The hollow one. (c) Both the same.

Problem 6P (II) A child rolls a ball on a level floor 3.5 m to another child. If the ball makes 12.0 revolutions, what is its diameter?

Mammals that depend on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–33). On the basis of rotational dynamics, explain why this distribution of mass is advantageous

Two wheels having the same radius and mass rotate at the same angular velocity (Fig. 8–38). One wheel is made with spokes so nearly all the mass is at the rim. The other is a solid disk. How do their rotational kinetic energies compare? (a) They are nearly the same. (b) The wheel with spokes has about twice the KE. (c) The wheel with spokes has higher KE, but not twice as high. (d) The solid wheel has about twice the KE. (e) The solid wheel has higher KE, but not twice as high.

Problem 7P (II) (a) A grinding wheel 0.35 m in diameter rotates at 2200 rpm. Calculate its angular velocity in rad s. (b) What are the linear speed and acceleration of a point on the edge of the grinding wheel?

This book has three symmetry axes through its center, all mutually perpendicular. The book’s moment of inertia would be smallest about which of the three? Explain.

A person stands on a platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is \(I_P\). The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia \(I_W\) and angular velocity \(\omega_W\). What will be the angular velocity \(\omega_P\) of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a \(60^{\circ}\) angle to the vertical, (c) vertically downward? (d) What will \(\omega_P\) be if the person reaches up and stops the wheel in part (a)? See Sections 8-8 and 8-9.

EXERCISE A In Example we found that the carousel, after , rotates at an angular velocity \(\omega=0.48\ \mathrm{rad} / \mathrm{s}\), and continues to do so after \(t=8.0\mathrm{\ s}\) because the acceleration ceased. What are the frequency and period of the carousel when rotating at this constant angular velocity \(\omega=0.48\ \mathrm{rad} / \mathrm{s}\)? ________________ Equation Transcription: Text Transcription: omega=0.48 rad/s t=8.0 s omega=0.48 rad/s

EXERCISE B Two forces (\(F_B=20\ \mathrm N\) and \(F_A=30\mathrm{\ N}\)) are applied to a meter stick which can rotate about its left end, Fig. Force \(\vec{F}_{B}\) is applied perpendicularly at the midpoint. Which force exerts the greater torque: \(F_A,\ F_B\), or both the same? When more than one torque acts on an object, the angular acceleration is found to be proportional to the net torque. If all the torques acting on an object tend to rotate it in the same direction about a fixed axis of rotation, the net torque is the sum of the torques. But if, say, one torque acts to rotate an object in one direction, and a second torque acts to rotate the object in the opposite direction, the net torque is the difference of the two torques. We normally assign a positive sign to torques that act to rotate the object counterclockwise (just as \(\theta\) is usually positive counterclockwise), and a negative sign to torques that act to rotate the object clockwise. ________________ Equation Transcription: Text Transcription: F_B=20 N F_A=30 N vector{F}_B F_A, F_B theta vector{F}_B vector{F}_A 30^o

Problem 8EC Return to the Chapter-Opening Question, page 198, and answer it again now. Try to explain why you may have answered differently the first time. CHAPTER-OPENING QUESTION A solid ball and a solid cylinder roll down a ramp. They both start from rest at the same time and place. Which gets to the bottom first? (a) They get there at the same time. (b) They get there at almost exactly the same time except for frictional differences. (c) The ball gets there first. (d) The cylinder gets there first. (e) Can’t tell without knowing the mass and radius of each.

Problem 8ED When a spinning figure skater pulls in her arms, her moment of inertia decreases; to conserve angular momentum, her angular velocity increases. Does her rotational kinetic energy also increase? If so, where does the energy come from?

EXERCISE E In Example 8–16, what if he moves the axis only \(90^{\circ}\) so it is horizontal? (a) The same direction and speed as above; (b) the same as above, but slower; (c) the opposite result. ________________ Equation Transcription: Text Transcription: 90^o

Problem 8EF Suppose you are standing on the edge of a large freely rotating turntable. If you walk toward the center, (a) the turntable slows down; (b) the turntable speeds up; (c) its rotation speed is unchanged; (d) you need to know the walking speed to answer.

If you used 1000 J of energy to throw a ball, would it travel faster if you threw the ball (ignoring air resistance) (a) so that it was also rotating? (b) so that it wasn’t rotating? (c) It makes no difference.

Problem 8P (II) A bicycle with tires 68 cm in diameter travels 9.2 km. How many revolutions do the wheels make?

Can the mass of a rigid object be considered concentrated at its CM for rotational motion? Explain.

A small solid sphere and a small thin hoop are rolling along a horizontal surface with the same translational speed when they encounter a 20° rising slope. If these two objects roll up the slope without slipping,which will rise farther up the slope? (a) The sphere. (b) The hoop. (c) Both the same. (d)More information about the objects’ mass and diameter is needed.

Problem 9P (II) Calculate the angular velocity (a) of a clock’s second hand, (b) its minute hand, and (c) its hour hand. State in rad s. (d) What is the angular acceleration in each case?

The moment of inertia of a rotating solid disk about an axis through its \(\mathrm{CM}\) is \(\frac{1}{2} M R^{2}\) (Fig. 8-20c). Suppose instead that a parallel axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller? Explain why. Equation Transcription: Text Transcription: CM 1/2 MR^2

A small mass \(m\) on a string is rotating without friction in a circle. The string is shortened by pulling it through the axis of rotation without any external torque, Fig. 8–39. What happens to the angular velocity of the object? (a) It increases. (b) It decreases. (c) It remains the same. Equation Transcription: Text Transcription: m

(II) A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–41). (a) What is the linear speed of a child seated 1.2 m from the center? (b) What is her acceleration (give components)?

Two inclines have the same height but make different angles with the horizontal. The same steel ball rolls without slipping down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

A small mass \(m\) on a string is rotating without friction in a circle. The string is shortened by pulling it through the axis of rotation without any external torque, Fig. 8–39. What happens to the tangential velocity of the object? (a) It increases. (b) It decreases. (c) It remains the same. Equation Transcription: Text Transcription: m

Problem 11P (II) What is the linear speed, due to the Earth’s rotation, of a point (a) on the equator, (b) on the Arctic Circle (latitude 66.5° N), and (c) at a latitude of 42.0° N?

Two spheres look identical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.

If there were a great migration of people toward the Earth’s equator, the length of the day would (a) increase because of conservation of angular momentum. (b) decrease because of conservation of angular momentum. (c) decrease because of conservation of energy. (d) increase because of conservation of energy. (e) remain unaffected.

Problem 12P (II) Calculate the angular velocity of the Earth (a) in its orbit around the Sun, and (b) about its axis.

A sphere and a cylinder have the same radius and the same mass. They start from rest at the top of an incline. (a) Which reaches the bottom first? (b) Which has the greater speed at the bottom? (c) Which has the greater total kinetic energy at the bottom? (d) Which has the greater rotational kinetic energy? Explain your answers.

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, your angular velocity will (a) increase. (b) decrease. (c) stay the same.

Why do tightrope walkers (Fig. 8–34) carry a long, narrow rod?

Problem 14P (II) A 61-cm-diameter wheel accelerates uniformly about its center from 120 rpm to 280 rpm in 4.0 s. Determine (a) its angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating.

We claim that momentum and angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain.

Can the diver of Fig. 8–28 do a somersault without having any initial rotation when she leaves the board? Explain.

(II) A turntable of radius \(R_1\) is turned by a circular rubber roller of radius \(R_2\) in contact with it at their outer edges. What is the ratio of their angular velocities, \(\omega_1/\omega_2\)?

When a motorcyclist leaves the ground on a jump and leaves the throttle on (so the rear wheel spins), why does the front of the cycle rise up?

Problem 17P (I) An automobile engine slows down from 3500 rpm to 1200 rpm in 2.5 s. Calculate (a) its angular acceleration, assumed constant, and (b) the total number of revolutions the engine makes in this time.

A shortstop may leap into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–35). Explain.

(I) A centrifuge accelerates uniformly from rest to 15,000 rpm in 240 s. Through how many revolutions did it turn in this time?

The angular velocity of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Problem 19P (I) Pilots can be tested for the stresses of flying high-speed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 23 complete revolutions before reaching its final speed. (a) What was its angular acceleration (assumed constant), and (b) what was its final angular speed in rpm?

In what direction is the Earth’s angular velocity vector as it rotates daily about its axis, north or south?

Problem 20P (II) A cooling fan is turned off when it is running at 850 rev/min. It turns 1250 revolutions before it comes to a stop. (a) What was the fan’s angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop?

On the basis of the law of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate in order to keep the helicopter stable.

(II) A wheel 31 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.8 s. How far will a point on the edge of the wheel have traveled in this time?

Problem 22P (II) The tires of a car make 75 revolutions as the car reduces its speed uniformly from 95 km/h to 55 km/h. The tires have a diameter of 0.80 m. (a) What was the angular acceleration of the tires? If the car continues to decelerate at this rate, (b) how much more time is required for it to stop, and (c) how far does it go?

Problem 23P (II) A small rubber wheel is used to drive a large pottery wheel. The two wheels are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of 7.2 rad/s2 and it is in contact with the pottery wheel (radius 27.0 cm) without slipping. Calculate (a) the angular acceleration of the pottery wheel, and (b) the time it takes the pottery wheel to reach its required speed of 65 rpm.

Problem 24P (I) A 52-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. (a) What is the maximum torque she exerts? (b) How could she exert more torque?

(II) Calculate the net torque about the axle of the wheel shown in Fig. 8–42. Assume that a friction torque of \(0.60\ \mathrm {m \cdot N}\) opposes the motion. ________________ Equation Transcription: Text Transcription: 0.60 m cdot N 135^o

(II) A person exerts a horizontal force of 42 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted (a) perpendicular to the door and (b) at a 60.0° angle to the face of the door?

(II) Two blocks, each of mass , are attached to the ends of a massless rod which pivots as shown in Fig. 8–43. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system when it is first released. ________________ Equation Transcription: ?1 ?2 Text Transcription: ell_1 ell_2

(II) The bolts on the cylinder head of an engine require tightening to a torque of \(95 \mathrm{\ m} \cdot \mathrm{N}\) If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm across (Fig. 8–44), estimate the force applied near each of the six points by a wrench. ________________ Equation Transcription: Text Transcription: 95 m cdot N vector{F}_on bolt vector{F}_on wrench

(II) Determine the net torque on the 2.0-m-long uniform beam shown in Fig. 8–45. All forces are shown. Calculate about (a) point C, the CM, and (b) point P at one end. FIGURE 8-45 Problem 29 ________________ Equation Transcription: Text Transcription: 32^o 45^o 58^o

Problem 30P (I) Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center.

(I) Estimate the moment of inertia of a bicycle wheel 67 cm in diameter. The rim and tire have a combined mass of 1.1 kg. The mass of the hub (at the center) can be ignored (why?).

Problem 32P (II) A merry-go-round accelerates from rest to 0.68 rad/s in 34 s. Assuming the merry-go-round is a uniform disk of radius 7.0 m and mass 31,000 kg, calculate the net torque required to accelerate it.

Problem 33P (II) An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 X 10-26 kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 X 10-49 kg.m2 From these data, estimate the effective distance between the atoms.

(II) The forearm in Fig. 8–46 accelerates a 3.6-kg ball at \(7.0 \mathrm{\ m} / \mathrm{s}^{2}\) by means of the triceps muscle, as shown. Calculate (a) the torque needed, and (b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm. ________________ Equation Transcription: Text Transcription: 7.0 m/s^2

(II) Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–46. The ball is accelerated uniformly from rest to \(\text {8.5 m/s}\) in 0.38 s, at which point it is released. Calculate (a) the angular acceleration of the arm, and (b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.7 kg and rotates like a uniform rod about an axis at its end. ________________ Equation Transcription: Text Transcription: 8.5 m/s

(II) A softball player swings a bat, accelerating it from rest to 2.6 rev/s in a time of 0.20 s. Approximate the bat as a 0.90-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.

Problem 38P (II) A small 350-gram ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculate (a) the moment of inertia of the ball about the center of the circle, and (b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore air resistance on the rod and its moment of inertia.

(II) Calculate the moment of inertia of the array of point objects shown in Fig. 8–47 about (a) the axis, and (b) the axis. Assume \(m=2.2 \mathrm{\ kg}\), \(M=3.4 \mathrm{\ kg}\), and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the axis. (c) About which axis would it be harder to accelerate this array? ________________ Equation Transcription: Text Transcription: m=2.2 kg M=3.4 kg

(II) A potter is shaping a bowl on a potter’s wheel rotating at constant angular velocity of 1.6 rev/s (Fig. 8–48). The friction force between her hands and the clay is 1.5 N total. (a) How large is her torque on the wheel, if the diameter of the bowl is 9.0 cm? (b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hands? The moment of inertia of the wheel and the bowl is \(0.11\ kg \cdot m^2\).

Problem 41P (II) A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 560 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?

Problem 42P (II) A 0.72-m-diameter solid sphere can be rotated about an axis through its center by a torque of 10.8 .m.N which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere?

(II) Let us treat a helicopter rotor blade as a long thin rod, as shown in Fig. 8–49. (a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. (b) How much torque must the motor apply to bring the blades from rest up to a speed of \(6.0\ \mathrm{rev} / \mathrm{s}\) in 8.0 s? ________________ Equation Transcription: Text Transcription: 6.0 rev/s m=135 kg

(II) A centrifuge rotor rotating at 9200 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of If the mass of the rotor is 3.10 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest, and how long will it take?

(II) To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–50. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 32 rpm in 5.0 min, starting from rest?

(III) Two blocks are connected by a light string passing over a pulley of radius and moment of inertia . The blocks move (towards the right) with an acceleration of \(1.00 \mathrm{\ m} / \mathrm{s}^{2}\) along their frictionless inclines (see Fig. 8-51). (a) Draw free-body diagrams for each of the two blocks and the pulley. ( ) Determine \(F_{T A}\) and \(F_{T B}\), the tensions in the two parts of the string. Find the net torque acting on the pulley, and determine its moment of inertia, . ________________ Equation Transcription: Text Transcription: 1.00 m/s^2 F_TA F_TB a=1.00 m/s^2 F_TA F_TB m_B=10.0 kg 61^o 32^o m_A=8.0 kg

Problem 47P An Atwood machine consists of two masses, \(m_{A}=65 \mathrm{\ kg}\) and \(m_{B}=75 \mathrm{\ kg}\), connected by a massless inelastic cord that passes over a pulley free to rotate, Fig. 8–52. The pulley is a solid cylinder of radius \(R=0.45 \mathrm{\ m}\) and mass 6.0 kg. (a) Determine the acceleration of each mass. (b) What % error would be made if the moment of inertia of the pulley is ignored? [Hint: The tensions \(F_{T A}\) and \(F_{T B}\) are not equal. We discussed the Atwood machine in Example 4–13, assuming \(I=0\)for the pulley.] Atwood machine. ________________ Equation Transcription: Text Transcription: m_A=65 kg m_B=75 kg R=0.45 m F_TA F_TB I=0 vector{F}_TA vector{F}_TB m_A m_B

Problem 49P (I) An automobile engine develops a torque of 265 m.N at 3350 rpm. What is the horsepower of the engine?

Problem 50P (I) A centrifuge rotor has a moment of inertia of 3.25 x 10-2 kg . m2. How much energy is required to bring it from rest to 8750 rpm?

Problem 51P (I) Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.20 m high. Assume it starts from rest and rolls without slipping.

Problem 52P (II) A bowling ball of mass 7.25 kg and radius 10.8 cm rolls without slipping down a lane at 3.10 m/s. Calculate its total kinetic energy.

Problem 53P (II) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (a) that due to its daily rotation about its axis, and (b) that due to its yearly revolution about the Sun. [Assume the Earth is a uniform sphere with mass 6.0 x 1024 kg, radius = 6.4 x 106 m, and is 1.5 x 108 km from the Sun.]

(II) A rotating uniform cylindrical platform of mass 220 kg and radius 5.5 m slows down from 3.8 rev/s to rest in 16 s when the driving motor is disconnected. Estimate the power output of the motor (hp) required to maintain a steady speed of 3.8 rev/s.

Problem 55P (II) A merry-go-round has a mass of 1440 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 7.00 s? Assume it is a solid cylinder.

Problem 56P (II) A sphere of radius r = 34.5 cm and mass m = 1.80 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long. (a) Calculate its translational and rotational speeds when it reaches the bottom. (b) What is the ratio of translational to rotational kinetic energy at the bottom? Avoid putting in numbers until the end so you can answer: (c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

Problem 57P A ball of radius r rolls on the inside of a track of radius R (see Fig. 8–53). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping? ________________ Equation Transcription: Text Transcription: 90^o

Problem 58P Two masses, \(m_{A}=32.0 \mathrm{\ kg}\) and \(m_B=38.0\mathrm{\ kg}\) are connected by a rope that hangs over a pulley (as in Fig. 8–54). The pulley is a uniform cylinder of radius \(R=0.311\ \mathrm m\) and mass 3.1 kg. Initially \(m_{A}\) is on the ground and \(m_{B}\) rests 2.5 m above the ground. If the system is released, use conservation of energy to determine the speed of \(m_{B}\) just before it strikes the ground. Assume the pulley bearing is frictionless. ________________ Equation Transcription: Text Transcription: m_A=32.0 kg m_B=38.0 kg R=0.311 m m_A m_B m_B m_A m_B

Problem 59P (III) A 1.80-m-long pole is balanced vertically with its tip on the ground. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]

Problem 60P (I) What is the angular momentum of a 0.270-kg ball revolving on the end of a thin string in a circle of radius 1.35 m at an angular speed of 10.4 rad/s?

Problem 62P A person stands, hands at his side, on a platform that is rotating at a rate of \(\text {0.90 rev/s}\). If he raises his arms to a horizontal position, Fig. 8–55, the speed of rotation decreases to \(\text {0.60 rev/s}\). (a) Why? (b) By what factor has his moment of inertia changed? ________________ Equation Transcription: Text Transcription: 0.90 rev/s 0.60 rev/s

Problem 63P II) A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed ?. Assuming no external torques, what is the final common angular speed of the two disks?

Problem 64P A diver (such as the one shown in Fig. 8–28) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed (\(\text {rev/s}\)) when in the straight position? ________________ Equation Transcription: Text Transcription: rev/s

Problem 66P (II) (a) What is the angular momentum of a figure skater spinning at 3.0 rev/s with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 48 kg? (b) How much torque is required to slow her to a stop in 4.0 s, assuming she does not move her arms?

Problem 67P (II) A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 820 kg . m2. The platform rotates without friction with angular velocity 0.95 rad/s. The person walks radially to the edge of the platform. (a) Calculate the angular velocity when the person reaches the edge. (b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.

Problem 68P (II) A potter’s wheel is rotating around a vertical axis through its center at a frequency of 1.5 rev/s. The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 2.6-kg chunk of clay, approximately shaped as a flat disk of radius 7.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it? Ignore friction.

Problem 69P (II) A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0.80 rad/s. Its total moment of inertia is 1360 kg . m2. Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. (a) What is the angular velocity of the merry-go-round now? (b) What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

(II) A uniform horizontal rod of mass M and length \(\ell\) rotates with angular velocity \(\omega\) about a vertical axis through its center. Attached to each end of the rod is a small mass m. Determine the angular momentum of the system about the axis.

Problem 71P (II) Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, (a) what would the Sun’s new rotation rate be? Take the Sun’s current period to be about 30 days. (b) What would be its final kinetic energy in terms of its initial kinetic energy of today?

(II) A uniform disk turns at 3.3 rev/s around a frictionless central axis. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–56. They then turn together around the axis with their centers superposed. What is the angular frequency in rev/s of the combination?

Problem 73P (III) An asteroid of mass 1.0 x 105 kg, traveling at a speed of 35 km/s relative to the Earth, hits the Earth at the equator tangentially, in the direction of Earth’s rotation, and is embedded there. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.

Problem 74P (III) Suppose a 65-kg person stands at the edge of a 5.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1850 kg . m2. The turntable is at rest initially, but when the person begins running at a speed of 4.0 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.

Problem 75GP A merry-go-round with a moment of inertia equal to 1260 kg . m2 and a radius of 2.5 m rotates with negligible friction at 1.70 rad/s. A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation, causing the platform to slow to 1.35 rad/s. What is her mass?

Problem 76GP A 1.6-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of 24 rev/s from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.

Problem 77GP On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius R1 = 2.5 cm and winds its way out to radius R2 = 5.8 cm. To read the digital information, a CD player rotates the CD so that the player’s readout laser scans along the spiral’s sequence of bits at a constant linear speed of 1.25 m/s Thus the player must accurately adjust the rotational frequency f of the CD as the laser moves outward. Determine the values for f (in units of rpm) when the laser is located at R1 and when it is at R2 .

(a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.013 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest. (b)What fraction of its kinetic energy is rotational?

Problem 79GP A cyclist accelerates from rest at a rate of \(1.00 \mathrm{\ m} / \mathrm{s}^{2}\). How fast will a point at the top of the rim of the tire (\(\text { diameter }=68.0 \mathrm{\ cm}\)) be moving after 2.25 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest—see Fig. 8–57.] ________________ Equation Transcription: Text Transcription: 1.00 m/s^2 diameter=68.0 cm v=? a=1.00 m/s^2

Problem 80GP Suppose David puts a 0.60-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle, accelerating it from rest to a rate of 75 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?

Bicycle gears: (a) How is the angular velocity \(\omega_R\) of the rear wheel of a bicycle related to the angular velocity \(\omega_F\) of the front sprocket and pedals? Let \(N_F\) and \(N_R\) be the number of teeth on the front and rear sprockets, respectively, Fig. 8–58. The teeth are spaced the same on both sprockets and the rear sprocket is firmly attached to the rear wheel. (b) Evaluate the ratio \(\omega_R / \omega_F\) when the front and rear sprockets have 52 and 13 teeth, respectively, and (c) when they have 42 and 28 teeth.

Figure 8–59 illustrates an \(H_2 O\) molecule. The O-H bond length is 0.096 nm and the H-O-H bonds make an angle of \(104^{\circ}\). Calculate the moment of inertia of the \(H_2 O\) molecule (assume the atoms are points) about an axis passing through the center of the oxygen atom (a) perpendicular to the plane of the molecule, and (b) in the plane of the molecule, bisecting the H-O-H bonds.

A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.0 m/s when it reaches a \(15^{\circ}\) incline. (a) How far up the incline will it go? (b) How long will it be on the incline before it arrives back at the bottom?

Determine the angular momentum of the Earth (a) about its rotation axis (assume the Earth is a uniform sphere), and (b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun).

Problem 86GP If the coefficient of static friction between a car’s tires and the pavement is 0.65, calculate the minimum torque that must be applied to the 66-cm-diameter tire of a 1080-kg automobile in order to “lay rubber” (make the wheels spin, slipping as the car accelerates). Assume each wheel supports an equal share of the weight

Problem 85GP A wheel of mass M has radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–60). The step has height h, where \(\text {h < R}\). What minimum force F is needed? ________________ Equation Transcription: Text Transcription: h<R vector{F}

Problem 87GP A 4.00-kg mass and a 3.00-kg mass are attached to opposite ends of a very light 42.0-cm-long horizontal rod (Fig. 8–61). The system is rotating at angular speed \(\omega=5.60\ \mathrm{rad} / \mathrm{s}\) about a vertical axle at the center of the rod. Determine (a) the kinetic energy KE of the system, and (b) the net force on each mass. ________________ Equation Transcription: Text Transcription: omega=5.60 rad/s vector{omega}

Problem 88GP A small mass m attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table (Fig. 8–62). Initially, the mass revolves with a speed \(v_{1}=2.4 \mathrm{\ m} / \mathrm{s}\) in a circle of radius \(r_{1}=0.80 \mathrm{\ m}\). The string is then pulled slowly through the hole so that the radius is reduced to \(r_{2}=0.48 \mathrm{\ m}\). What is the speed, \(v_{2}\), of the mass now? ________________ Equation Transcription: Text Transcription: v_1=2.4 m/s r_1=0.80 m r_2=0.48 m v_2 r_1 v_1

A uniform rod of mass M and length \(\ell\) can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–63. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–20g.]

Problem 90GP Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing \(\frac{3}{4}\) of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrownoff mass carries off either (a) no angular momentum, or (b) its proportional share \(\left(\frac{3}{4}\right)\) of the initial angular momentum. ________________ Equation Transcription: Text Transcription: 3 over 4 (3 over 4)

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance \(\ell\), holding onto it, Fig. 8–64. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

Problem 92GP The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

Problem 93GP A spherical asteroid with radius \(r=123 \mathrm{~m}\) and mass \(M=2.25 \times 10^{10} \mathrm{\ kg}\) rotates about an axis at four revolutions per day. A “tug” spaceship attaches itself to the asteroid’s south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the asteroid’s surface as shown in Fig. 8–65. If \(F=285 \mathrm{~N}\), how long will it take the tug to rotate the asteroid’s axis of rotation through an angle of \(5.0^{\circ}\) by this method? ________________ Equation Transcription: Text Transcription: r=123 m M=2.25x10^10 kg F=285 N 5.0^o r=123 m F=285 N

Problem 94GP Most of our Solar System’s mass is contained in the Sun, and the planets possess almost all of the Solar System’s angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar System’s total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass \(1.99\times10^{30}\mathrm{\ kg}\), radius \(6.96\times10^8\mathrm{\ m}\)) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planet’s spin about its own axis. Planet Mean Distance from Sun (\(\times10^6\mathrm{\ km}\)) Orbital Period (Earth Years) Mass (\(\times10^{25}\mathrm{\ kg}\)) Jupiter 778 11.9 190 Saturn 1427 29.5 56.8 Uranus 2870 84.0 8.68 Neptune 4500 165 10.2 ________________ Equation Transcription: Text Transcription: 1.99x10^30 kg 6.96x10^8 m x10^6 km x10^25 kg

The radius of the roll of paper shown in Fig. 8–67 is 7.6 cm and its moment of inertia is \(I = 3.3 \times 10^{-3}\ kg \cdot m^2\). A force of 3.5 N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque of \(0.11\ m \cdot N\) is exerted on the roll which gradually brings it to a stop. Assuming that the paper’s thickness is negligible, calculate (a) the length of paper that unrolls during the time that the force is applied (1.3 s) and (b) the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving.

Problem 96GP The radius of the roll of paper shown in Fig. 8–67 is 7.6 cm and its moment of inertia is I = 3.3 X 10-3 kg.m2 . A force of 3.5 N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque 0.11 m.N of is exerted on the roll which gradually brings it to a stop. Assuming that the paper’s thickness is negligible, calculate (a) the length of paper that unrolls during the time that the force is applied (1.3 s) and (b) the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving.

Express the following angles in radians: (a) 45.0, (b) 60.0, (c) 90.0, (d) 360.0, and (e) 445. Give as numerical values and as fractions of

The Sun subtends an angle of about 0.5 to us on Earth, 150 million km away. Estimate the radius of the Sun.

A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle (Fig. 840) of What diameter spot will it make on the Moon?

The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 4.0 s. What is the angular acceleration as the blades slow down?

The platter of the hard drive of a computer rotates at per minute (a) What is the angular velocity of the platter? (b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? (c) If a single bit requires of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

A child rolls a ball on a level floor 3.5 m to another child. If the ball makes 12.0 revolutions, what is its diameter?

A grinding wheel 0.35 m in diameter rotates at 2200 rpm. Calculate its angular velocity in rad s. (b) What are the linear speed and acceleration of a point on the edge of the grinding wheel?

A bicycle with tires 68 cm in diameter travels 9.2 km. How many revolutions do the wheels make?

(II) Calculate the angular velocity (a) of a clock’s second hand, (b) its minute hand, and (c) its hour hand. State in rad/s. (d) What is the angular acceleration in each case?

A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 841). (a) What is the linear speed of a child seated 1.2 m from the center? (b) What is her acceleration (give components)?

What is the linear speed, due to the Earths rotation, of a point (a) on the equator, (b) on the Arctic Circle (latitude 66.5 N), and (c) at a latitude of 42.0 N?

Calculate the angular velocity of the Earth (a) in its orbit around the Sun, and (b) about its axi

How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 gs?

A 61-cm-diameter wheel accelerates uniformly about its center from 120 rpm to 280 rpm in 4.0 s. Determine (a) its angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating.

In traveling to the Moon, astronauts aboard the Apollo spacecraft put the spacecraft into a slow rotation to distribute the Suns energy evenly (so one side would not become too hot). At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. Think of the spacecraft as a cylinder with a diameter of 8.5 m rotating about its cylindrical axis. Determine (a) the angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 6.0 min after it started this acceleration.

(II) A turntable of radius \(R_1\) is turned by a circular rubber roller of radius \(R_2\) in contact with it at their outer edges. What is the ratio of their angular velocities, \(\omega_1/\omega_2\)?

An automobile engine slows down from 3500 rpm to 1200 rpm in 2.5 s. Calculate (a) its angular acceleration, assumed constant, and (b) the total number of revolutions the engine makes in this time

A centrifuge accelerates uniformly from rest to 15,000 rpm in 240 s. Through how many revolutions did it turn in this time?

Pilots can be tested for the stresses of flying high-speed jets in a whirling human centrifuge, which takes 1.0 min to turn through 23 complete revolutions before reaching its final speed. (a) What was its angular acceleration (assumed constant), and (b) what was its final angular speed in rpm?

A cooling fan is turned off when it is running at It turns 1250 revolutions before it comes to a stop. (a) What was the fans angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop?

A wheel 31 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.8 s. How far will a point on the edge of the wheel have traveled in this time?

(II) The tires of a car make 75 revolutions as the car reduces its speed uniformly from 95 km/h to 55 km/h. The tires have a diameter of 0.80 m. (a) What was the angular acceleration of the tires? If the car continues to decelerate at this rate, (b) how much more time is required for it to stop, and (c) how far does it go?

A small rubber wheel is used to drive a large pottery wheel. The two wheels are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of and it is in contact with the pottery wheel (radius 27.0 cm) without slipping. Calculate (a) the angular acceleration of the pottery wheel, and (b) the time it takes the pottery wheel to reach its required speed of 65 rpm.

A 52-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. (a) What is the maximum torque she exerts? (b) How could she exert more torque?

Calculate the net torque about the axle of the wheel shown in Fig. 842. Assume that a friction torque of 0.60 mN opposes the motion.

A person exerts a horizontal force of 42 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted (a) perpendicular to the door and (b) at a 60.0 angle to the face of the door?

Two blocks, each of mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 843. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system when it is first released

The bolts on the cylinder head of an engine require tightening to a torque of If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm across (Fig. 844), estimate the force applied near each of the six points by a wrench.

(II) Determine the net torque on the 2.0-m-long uniform beam shown in Fig. 8-45. All forces are shown. Calculate about (a) point C, the \(\mathrm{CM}\), and (b) point P at one end. Equation Transcription: Text Transcription: CM

Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center

Estimate the moment of inertia of a bicycle wheel 67 cm in diameter. The rim and tire have a combined mass of 1.1 kg. The mass of the hub (at the center) can be ignored (why?).

A merry-go-round accelerates from rest to in 34 s. Assuming the merry-go-round is a uniform disk of radius 7.0 m and mass 31,000 kg, calculate the net torque required to accelerate it.

(II) An oxygen molecule consists of two oxygen atoms whose total mass is \(5.3 \times 10^{-26}\ kg\) and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is \(1.9 \times 10^{-46}\ kg \cdot m^2\). From these data, estimate the effective distance between the atoms.

A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.380 kg. Calculate (a) its moment of inertia about its center, and (b) the applied torque needed to accelerate it from rest to 1750 rpm in 5.00 s. Take into account a frictional torque that has been measured to slow down the wheel from 1500 rpm to rest in 55.0 s.

The forearm in Fig. 846 accelerates a 3.6-kg ball at by means of the triceps muscle, as shown. Calculate (a) the torque needed, and (b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm

Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 846. The ball is accelerated uniformly from rest to in 0.38 s, at which point it is released. Calculate (a) the angular acceleration of the arm, and (b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.7 kg and rotates like a uniform rod about an axis at its end

A softball player swings a bat, accelerating it from rest to in a time of 0.20 s. Approximate the bat as a 0.90-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it

A small 350-gram ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculate (a) the moment of inertia of the ball about the center of the circle, and (b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore air resistance on the rod and its moment of inertia.

(II) Calculate the moment of inertia of the array of point objects shown in Fig. 8–47 about (a) the y axis, and (b) the x axis. Assume m = 2.2 kg, M = 3.4 kg, and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the x axis. (c) About which axis would it be harder to accelerate this array?

A potter is shaping a bowl on a potters wheel rotating at constant angular velocity of (Fig. 848). The friction force between her hands and the clay is 1.5 N total. (a) How large is her torque on the wheel, if the diameter of the bowl is 9.0 cm? (b) How long would it take for the potters wheel to stop if the only torque acting on it is due to the potters hands? The moment of inertia of the wheel and the bowl is 0.11 kgm2 .

A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 560 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?

A 0.72-m-diameter solid sphere can be rotated about an axis through its center by a torque of which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere?

Let us treat a helicopter rotor blade as a long thin rod, as shown in Fig. 849. (a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. (b) How much torque must the motor apply to bring the blades from rest up to a speed of in 8.0 s?

A centrifuge rotor rotating at 9200 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of If the mass of the rotor is 3.10 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest, and how long will it take?

To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 850. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 32 rpm in 5.0 min, starting from rest?

Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia I. The blocks move (towards the right) with an acceleration of along their frictionless inclines (see Fig. 851). (a) Draw free-body diagrams for each of the two blocks and the pulley. (b) Determine and the tensions in the two parts of the string. (c) Find the net torque acting on the pulley, and determine its moment of inertia, I.

An Atwood machine consists of two masses, and connected by a massless inelastic cord that passes over a pulley free to rotate, Fig. 852. The pulley is a solid cylinder of radius and mass 6.0 kg. (a) Determine the acceleration of each mass. (b) What % error would be made if the moment of inertia of the pulley is ignored? [Hint: The tensions and are not equal. We discussed the Atwood machine in Example 413, assuming I = 0 for the pulley.]

A hammer thrower accelerates the hammer from rest within four full turns (revolutions) and releases it at a speed of Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate (a) the angular acceleration, (b) the (linear) tangential acceleration, (c) the centripetal acceleration just before release, (d) the net force being exerted on the hammer by the athlete just before release, and (e) the angle of this force with respect to the radius of the circular motion. Ignore gravity

An automobile engine develops a torque of at 3350 rpm. What is the horsepower of the engine?

A centrifuge rotor has a moment of inertia of How much energy is required to bring it from rest to 8750 rpm?

Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.20 m high. Assume it starts from rest and rolls without slipping

A bowling ball of mass 7.25 kg and radius 10.8 cm rolls without slipping down a lane at Calculate its total kinetic energy

Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (a) that due to its daily rotation about its axis, and (b) that due to its yearly revolution about the Sun. [Assume the Earth is a uniform sphere with and is

A rotating uniform cylindrical platform of mass 220 kg and radius 5.5 m slows down from to rest in 16 s when the driving motor is disconnected. Estimate the power output of the motor (hp) required to maintain a steady speed of 3.8 rev/s?

A merry-go-round has a mass of 1440 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 7.00 s? Assume it is a solid cylinder

A sphere of radius and mass starts from rest and rolls without slipping down a 30.0 incline that is 10.0 m long. (a) Calculate its translational and rotational speeds when it reaches the bottom. (b) What is the ratio of translational to rotational kinetic energy at the bottom? Avoid putting in numbers until the end so you can answer: (c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

A ball of radius r rolls on the inside of a track of radius R (see Fig. 853). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping?

Two masses, and are connected by a rope that hangs over a pulley (as in Fig. 854). The pulley is a uniform cylinder of radius and mass 3.1 kg. Initially is on the ground and rests 2.5 m above the ground. If the system is released, use conservation of energy to determine the speed of just before it strikes the ground. Assume the pulley bearing is frictionless.

A 1.80-m-long pole is balanced vertically with its tip on the ground. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.

(I) What is the angular momentum of a 0.270-kg ball revolving on the end of a thin string in a circle of radius 1.35 m at an angular speed of 10.4 rad/s?

(I) (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 28 cm when rotating at 1300 rpm? (b) How much torque is required to stop it in 6.0 s?

A person stands, hands at his side, on a platform that is rotating at a rate of If he raises his arms to a horizontal position, Fig. 855, the speed of rotation decreases to (a) Why? (b) By what factor has his moment of inertia changed?

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed Assuming no external torques, what is the final common angular speed of the two disks?

A diver (such as the one shown in Fig. 828) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed (rev s) when in the straight position?

A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of If her initial moment of inertia was what is her final moment of inertia? How does she physically accomplish this change?

(a) What is the angular momentum of a figure skater spinning at with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 48 kg? (b) How much torque is required to slow her to a stop in 4.0 s, assuming she does not move her arms?

A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia The platform rotates without friction with angular velocity The person walks radially to the edge of the platform. (a) Calculate the angular velocity when the person reaches the edge. (b) Calculate the rotational kinetic energy of the system of platform plus person before and after the persons walk.

(II) A potter’s wheel is rotating around a vertical axis through its center at a frequency of 1.5 rev/s. The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 2.6-kg chunk of clay, approximately shaped as a flat disk of radius 7.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it? Ignore friction.

A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of Its total moment of inertia is Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. (a) What is the angular velocity of the merry-go-round now? (b) What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

A uniform horizontal rod of mass M and length rotates with angular velocity about a vertical axis through its center. Attached to each end of the rod is a small mass m. Determine the angular momentum of the system about the axis.

Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, (a) what would the Suns new rotation rate be? Take the Suns current period to be about 30 days. (b) What would be its final kinetic energy in terms of its initial kinetic energy of today?

(II) A uniform disk turns at 3.3 rev/s around a frictionless central axis. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–56. They then turn together around the axis with their centers superposed. What is the angular frequency in rev/s of the combination?

An asteroid of mass traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, in the direction of Earths rotation, and is embedded there. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.

Suppose a 65-kg person stands at the edge of a 5.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of The turntable is at rest initially, but when the person begins running at a speed of (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.

A merry-go-round with a moment of inertia equal to and a radius of 2.5 m rotates with negligible friction at A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation, causing the platform to slow to What is her mass?

A 1.6-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.

On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius and winds its way out to radius To read the digital information, a CD player rotates the CD so that the players readout laser scans along the spirals sequence of bits at a constant linear speed of Thus the player must accurately adjust the rotational frequency f of the CD as the laser moves outward. Determine the values for f (in units of rpm) when the laser is located at and when it is at R2 R .

(a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.013 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest. (b) What fraction of its kinetic energy is rotational?

A cyclist accelerates from rest at a rate of How fast will a point at the top of the rim of the tire be moving after 2.25 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at restsee Fig. 857.]

Suppose David puts a 0.60-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle, accelerating it from rest to a rate of 75 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?

Bicycle gears: (a) How is the angular velocity of the rear wheel of a bicycle related to the angular velocity of the front sprocket and pedals? Let and be the number of teeth on the front and rear sprockets, respectively, Fig. 858. The teeth are spaced the same on both sprockets and the rear sprocket is firmly attached to the rear wheel. (b) Evaluate the ratio when the front and rear sprockets have 52 and 13 teeth, respectively, and (c) when they have 42 and 28 teeth.

Figure 859 illustrates an molecule. The bond length is 0.096 nm and the bonds make an angle of 104. Calculate the moment of inertia of the molecule (assume the atoms are points) about an axis passing through the center of the oxygen atom (a) perpendicular to the plane of the molecule, and (b) in the plane of the molecule, bisecting the bonds.

A hollow cylinder (hoop) is rolling on a horizontal surface at speed when it reaches a 15 incline. (a) How far up the incline will it go? (b) How long will it be on the incline before it arrives back at the bottom?

Determine the angular momentum of the Earth (a) about its rotation axis (assume the Earth is a uniform sphere), and (b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun)

A wheel of mass M has radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 860). The step has height h, where What minimum force F is needed?

If the coefficient of static friction between a cars tires and the pavement is 0.65, calculate the minimum torque that must be applied to the 66-cm-diameter tire of a 1080-kg automobile in order to lay rubber (make the wheels spin, slipping as the car accelerates). Assume each wheel supports an equal share of the weight

A 4.00-kg mass and a 3.00-kg mass are attached to opposite ends of a very light 42.0-cm-long horizontal rod (Fig. 861). The system is rotating at angular speed about a vertical axle at the center of the rod. Determine (a) the kinetic energy KE of the system, and (b) the net force on each mass

A small mass m attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table (Fig. 862). Initially, the mass revolves with a speed in a circle of radius The string is then pulled slowly through the hole so that the radius is reduced to What is the speed, of the mass now?

A uniform rod of mass M and length can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 863. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 820g.]

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrownoff mass carries off either (a) no angular momentum, or (b) its proportional share of the initial angular momentum.

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance , holding onto it, Fig. 864. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spools center of mass move?

The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

A spherical asteroid with radius r = 123 m and mass \(M = 2.25 \times 10^{10}\ kg\) rotates about an axis at four revolutions per day. A “tug” spaceship attaches itself to the asteroid’s south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the asteroid’s surface as shown in Fig. 8–65. If F = 285 N, how long will it take the tug to rotate the asteroid’s axis of rotation through an angle of \(5.0^{\circ}\) by this method?

Most of our Solar Systems mass is contained in the Sun, and the planets possess almost all of the Solar Systems angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar Systems total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass radius ) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planets spin about its own axis.

Water drives a waterwheel (or turbine) of radius as shown in Fig. 866. The water enters at a speed and exits from the waterwheel at a speed (a) If 85 kg of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5 s, how much power is delivered to the wheel?

The radius of the roll of paper shown in Fig. 8–67 is 7.6 cm and its moment of inertia is \(I = 3.3 \times 10^{-3}\ kg.m^2\). A force of 3.5N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque of \(0.11 m \cdot N\) is exerted on the roll which gradually brings it to a stop. Assuming that the paper’s thickness is negligible, calculate (a) the length of paper that unrolls during the time that the force is applied (1.3 s) and (b) the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving.