A persons center of mass is easily found by having the

What is a rigid body?

What is torque?

What does torque do?

What is angular momentum?

What is conserved in rotational motion?

Why is rigid-body motion important?

A barbell consists of a 500 g ball and a 2.0 kg ball connected by a massless 50-cm-long rod. a. Where is the center of mass? b. What is the speed of each ball if they rotate about the center of mass at 40 rpm?

A baseball bat is cut into two pieces at its center of mass. Which end is heavier? a. The handle end (left end). b. The hitting end (right end). c. The two ends weigh the same

Find the center of mass of a thin, uniform rod of length L and mass M. Use this result to find the tangential acceleration of one tip of a 1.60-m-long rod that rotates about its center of mass with an angular acceleration of 6.0 rad/s2 .

A solid cylinder and a cylindrical shell, each with radius R and mass M, rotate about their axes with the same angular velocity v. Which has more kinetic energy? a. The solid cylinder. b. The cylindrical shell. c. They have the same kinetic energy. d. Neither has kinetic energy because they are only rotating, not moving

Students participating in an engineering project design the triangular widget seen in FIGURE 12.10. The three masses, held together by lightweight plastic rods, rotate in the plane of the page about an axle passing through the right-angle corner. At what angular velocity does the widget have 100 mJ of rotational energy?

Four Ts are made from two identical rods of equal mass and length. Rank in order, from largest to smallest, the moments of inertia Ia to Id for rotation about the dashed line. (a) (b) (c) (d)

A 1.0-m-long, 200 g rod is hinged at one end and connected to a wall. It is held out horizontally, then released. What is the speed of the tip of the rod as it hits the wall?

Rank in order, from largest to smallest, the five torques ta to te. The rods all have the same length and are pivoted at the dot. 2 N 2 N 4 N 2 N 4 N 45 (a) (b) (c) (d) (e)

Find the moment of inertia of a thin, uniform rod of length L and mass M that rotates about a pivot at one end.

Rank in order, from largest to smallest, the angular accelerations aa to ad. 2 m (a) 2 kg 2 kg 1 N 1 N 4 m 2 kg (d) 2 kg 1 N 1 N 2 m (b) 2 kg 2 N 2 N 2 kg 2 m (c) 4 kg 1 N 1 N 4

Find the moment of inertia of a circular disk of radius R and mass M that rotates on an axis passing through its center

What does the scale read? a. 500 N b. 1000 N c. 2000 N d. 4000 N

Find the moment of inertia of a thin rod with mass M and length L about an axis one-third of the length from one end

Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, it starts to rain. As a result, a. The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane. b. The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved. c. The buckets speed up because the potential energy of the rain is transformed into kinetic energy. d. The buckets slow down because the angular momentum of the bucket + rain system is conserved. e. Both a and b. f. None of the above.

Luis uses a 20-cm-long wrench to turn a nut. The wrench handle is tilted 30 above the horizontal, and Luis pulls straight down on the end with a force of 100 N. How much torque does Luis exert on the nut?

The 4.00-m-long, 500 kg steel beam shown in FIGURE 12.25 is supported 1.20 m from the right end. What is the gravitational torque about the support?

Far out in space, a 100,000 kg rocket and a 200,000 kg rocket are docked at opposite ends of a motionless 90-m-long connecting tunnel. The tunnel is rigid and its mass is much less than that of either rocket. The rockets start their engines simultaneously, each generating 50,000 N of thrust in opposite directions. What is the structures angular velocity after 30 s?

The engine in a small airplane is specified to have a torque of 60 Nm. This engine drives a 2.0-m-long, 40 kg propeller. On startup, how long does it take the propeller to reach 200 rpm?

A 2.0 kg bucket is attached to a massless string that is wrapped around a 1.0 kg, 4.0-cm-diameter cylinder, as shown in FIGURE 12.30a. The cylinder rotates on an axle through the center. The bucket is released from rest 1.0 m above the floor. How long does it take to reach the floor?

Weightlifting can exert extremely large forces on the bodys joints and tendons. In the strict curl event, a standing athlete uses both arms to lift a barbell by moving only his forearms, which pivot at the elbows. The record weight lifted in the strict curl is over 200 pounds (about 900 N). FIGURE 12.31 shows the arm bones and the biceps, the main lifting muscle when the forearm is horizontal. What is the tension in the tendon connecting the biceps muscle to the bone while a 900 N barbell is held stationary in this position?

Adrienne (50 kg) and Bo (90 kg) are playing on a 100 kg rigid plank resting on the supports seen in FIGURE 12.33. If Adrienne stands on the left end, can Bo walk all the way to the right end without the plank tipping over? If not, how far can he get past the support on the right?

A 3.0-m-long ladder leans against a frictionless wall at an angle of 60. What is the minimum value of ms, the coefficient of static friction with the ground, that prevents the ladder from slipping?

Figure 12.47 shows vectors C u and D u in the plane of the page. What is the cross product E u = C u * D u ?

Example 12.8 found the torque that Luis exerts on a nut by pulling on the end of a wrench. What is the torque vector?

Two equal masses are at the ends of a massless 50-cm-long rod. The rod spins at 2.0 rev/s about an axis through its midpoint. Suddenly, a compressed gas expands the rod out to a length of 160 cm. What is the rotation frequency after the expansion?

A 20-cm-diameter, 2.0 kg solid disk is rotating at 200 rpm. A 20-cm-diameter, 1.0 kg circular loop is dropped straight down onto the rotating disk. Friction causes the loop to accelerate until it is riding on the disk. What is the final angular velocity of the combined system?

A gyroscope used in a lecture demonstration consists of a 120 g, 7.0-cm-diameter solid disk that rotates on a lightweight axle. From the center of the disk to the end of the axle is 5.0 cm. When spun, placed on a stand, and released, the gyroscope is observed to precess with a period of 1.0 s. How fast, in rpm, is it spinning?

A 2.0 kg block hangs from the end of a 1.5 kg,1.0@m@long rod, together forming a pendulum that swings from a frictionless pivot at the top end of the rod. A 10 g bullet is fired horizontally into the block, where it sticks, causing the pendulum to swing out to a 30 angle. What was the speed of the bullet?

Is the center of mass of the dumbbell in FIGURE Q12.1 at point a, b, or c? Explain.

If the angular velocity v is held constant, by what factor must R change to double the rotational kinetic energy of the dumbbell in FIGURE Q12.2?

FIGURE Q12.3 shows three rotating disks, all of equal mass. Rank in order, from largest to smallest, their rotational kinetic energies Ka to Kc.

Must an object be rotating to have a moment of inertia? Explain

The moment of inertia of a uniform rod about an axis through its center is 1 12 mL2 . The moment of inertia about an axis at one end is 1 3 mL2 . Explain why the moment of inertia is larger about the end than about the center

You have two solid steel spheres. Sphere 2 has twice the radius of sphere 1. By what factor does the moment of inertia I2 of sphere 2 exceed the moment of inertia I1 of sphere 1?

The professor hands you two spheres. They have the same mass, the same radius, and the same exterior surface. The professor claims that one is a solid sphere and the other is hollow. Can you determine which is which without cutting them open? If so, how? If not, why not?

Six forces are applied to the door in FIGURE Q12.8. Rank in order, from largest to smallest, the six torques ta to tf about the hinge on the left. Explain.

A student gives a quick push to a ball at the end of a massless, rigid rod, as shown in FIGURE Q12.9, causing the ball to rotate clockwise in a horizontal circle. The rods pivot is frictionless. a. As the student is pushing, is the torque about the pivot positive, negative, or zero? b. After the push has ended, does the balls angular velocity (i) steadily increase; (ii) increase for awhile, then hold steady; (iii) hold steady; (iv) decrease for awhile, then hold steady; or (v) steadily decrease? Explain. c. Right after the push has ended, is the torque positive, negative, or zero?

Rank in order, from largest to smallest, the angular accelerations aa to ad in FIGURE Q12.10. Explain.

The solid cylinder and cylindrical shell in FIGURE Q12.11 have the same mass, same radius, and turn on frictionless, horizontal axles. (The cylindrical shell has lightweight spokes connecting the shell to the axle.) A rope is wrapped around each cylinder and tied to a block. The blocks have the same mass and are held the same height above the ground. Both blocks are released simultaneously. Which hits the ground first? Or is it a tie? Explain

A diver in the pike position (legs straight, hands on ankles) usually makes only one or one-and-a-half rotations. To make two or three rotations, the diver goes into a tuck position (knees bent, body curled up tight). Why?

Is the angular momentum of disk a in FIGURE Q12.13 larger than, smaller than, or equal to the angular momentum of disk b? Explain.

A high-speed drill reaches 2000 rpm in 0.50 s. a. What is the drills angular acceleration? b. Through how many revolutions does it turn during this first 0.50 s?

A skater holds her arms outstretched as she spins at 180 rpm. What is the speed of her hands if they are 140 cm apart?

A ceiling fan with 80-cm-diameter blades is turning at 60 rpm. Suppose the fan coasts to a stop 25 s after being turned off. a. What is the speed of the tip of a blade 10 s after the fan is turned off? b. Through how many revolutions does the fan turn while stopping?

An 18-cm-long bicycle crank arm, with a pedal at one end, is attached to a 20-cm-diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rate from 60 rpm to 90 rpm in 10 s. a. What is the tangential acceleration of the pedal? b. What length of chain passes over the top of the sprocket during this interval?

How far from the center of the earth is the center of mass of the earth + moon system? Data for the earth and moon can be found inside the back cover of the book.

The three masses shown in Figure EX12.6 are connected by massless, rigid rods. What are the coordinates of the center of mass?

The three masses shown in Figure EX12.7 are connected by massless, rigid rods. What are the coordinates of the center of mass?

A 100 g ball and a 200 g ball are connected by a 30-cm-long, massless, rigid rod. The balls rotate about their center of mass at 120 rpm. What is the speed of the 100 g ball?

A thin, 100 g disk with a diameter of 8.0 cm rotates about an axis through its center with 0.15 J of kinetic energy. What is the speed of a point on the rim?

What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth can be found inside the back cover of the book.

The three 200 g masses in Figure EX12.11 are connected by massless, rigid rods. a. What is the triangles moment of inertia about the axis through the center? b. What is the triangles kinetic energy if it rotates about the axis at 5.0 rev/s?

A drum major twirls a 96-cm-long, 400 g baton about its center of mass at 100 rpm. What is the batons rotational kinetic energy?

The four masses shown in Figure EX12.13 are connected by massless, rigid rods. a. Find the coordinates of the center of mass. b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page

The four masses shown in Figure EX12.13 are connected by massless, rigid rods. a. Find the coordinates of the center of mass. b. Find the moment of inertia about a diagonal axis that passes through masses B and D.

The three masses shown in Figure EX12.15 are connected by massless, rigid rods. a. Find the coordinates of the center of mass. b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page. c. Find the moment of inertia about an axis that passes through masses B and C.

A 12-cm-diameter CD has a mass of 21 g. What is the CDs moment of inertia for rotation about a perpendicular axis (a) through its center and (b) through the edge of the disk?

A 25 kg solid door is 220 cm tall, 91 cm wide. What is the doors moment of inertia for (a) rotation on its hinges and (b) rotation about a vertical axis inside the door, 15 cm from one edge?

In Figure EX12.18, what is the net torque about the axle?

Does ginkgo improve memory? The law allows marketers of herbs and other natural substances to make health claims that are not supported by evidence. Brands of ginkgo extract claim to improve memory and concentration. A randomized comparative experiment found no evidence for such effects.3 The subjects were 230 healthy people over 60 years old. They were randomly assigned to ginkgo or a placebo pill (a dummy pill that looks and tastes the same). All the subjects took a battery of tests for learning and memory before treatment started and again after six weeks. (a) Following the model of Figure 9.3, outline the design of this experiment. (b) Use the Simple Random Sample applet, other software, or Table B to assign half the subjects to the ginkgo group. If you use software, report the first 20 members of the ginkgo group (in the applets Sample bin) and the first 20 members of the placebo group (those left in the Population hopper). If you use Table B, start at line 103 and choose only the first 5 members of the ginkgo group.

The 20-cm-diameter disk in Figure EX12.20 can rotate on an axle through its center. What is the net torque about the axle?

A 5.00 kg particle starts from the origin at the time zero. its velocity as function of time is where v in meters per second and t is in seconds. (a) find its position as a function of time. (b) describe its motion qualitatively. Find (c) its acceleration as a function of time, (d) the net force exerted on the particle as a function of time, (e) the net torque about the origin exerted on the particle as a function of time,(f) the angular momentum of the particle as a function of time,(g) the kinetic energy from the particle as a function of time, and (h) the power injected into the system of the particle as a function of time.

A 4.0-m-long, 500 kg steel beam extends horizontally from the point where it has been bolted to the framework of a new building under construction. A 70 kg construction worker stands at the far end of the beam. What is the magnitude of the torque about the bolt due to the worker and the weight of the beam?

An athlete at the gym holds a 3.0 kg steel ball in his hand. His arm is 70 cm long and has a mass of 3.8 kg, with the center of mass at 40% of the arm length. What is the magnitude of the torque about his shoulder due to the ball and the weight of his arm if he holds his arm a. Straight out to his side, parallel to the floor? b. Straight, but 45 below horizontal?

An objects moment of inertia is 2.0 kgm2 . Its angular velocity is increasing at the rate of 4.0 rad/s per second. What is the net torque on the object?

An object whose moment of inertia is 4.0 kgm2 experiences the torque shown in Figure EX12.25. What is the objects angular velocity at t = 3.0 s? Assume it starts from rest

A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid, massless rod. The rod is rotating cw about its center of mass at 20 rpm. What net torque will bring the balls to a halt in 5.0 s?

Starting from rest, a 12-cm-diameter compact disk takes 3.0 s to reach its operating angular velocity of 2000 rpm. Assume that the angular acceleration is constant. The disks moment of inertia is 2.5 * 10-5 kgm2 . a. How much net torque is applied to the disk? b. How many revolutions does it make before reaching full speed?

A 4.0 kg, 36-cm-diameter metal disk, initially at rest, can rotate on an axle along its axis. A steady 5.0 N tangential force is applied to the edge of the disk. What is the disks angular velocity, in rpm, 4.0 s later

The two objects in Figure EX12.29 are balanced on the pivot. What is distance d?

The object shown in Figure EX12.30 is in equilibrium. What are the magnitudes of F u 1 and F u 2?

The 3.0-m-long, 100 kg rigid beam of Figure EX12.31 is supported at each end. An 80 kg student stands 2.0 m from support 1. How much upward force does each support exert on the beam?

A 5.0 kg cat and a 2.0 kg bowl of tuna fish are at opposite ends of the 4.0-m-long seesaw of Figure EX12.32. How far to the left of the pivot must a 4.0 kg cat stand to keep the seesaw balanced?

A car tire is 60 cm in diameter. The car is traveling at a speed of 20 m/s. a. What is the tires angular velocity, in rpm? b. What is the speed of a point at the top edge of the tire? c. What is the speed of a point at the bottom edge of the tire?

A 500 g, 8.0-cm-diameter can is filled with uniform, dense food. It rolls across the floor at 1.0 m/s. What is the cans kinetic energy?

An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom. a. What is the spheres angular velocity at the bottom of the incline? b. What fraction of its kinetic energy is rotational?

A solid sphere of radius R is placed at a height of 30 cm on a 15 slope. It is released and rolls, without slipping, to the bottom. From what height should a circular hoop of radius R be released on the same slope in order to equal the spheres speed at the bottom?

Evaluate the cross products A u * B u and C u * D u .

Evaluate the cross products A u * B u and C u * D u .

Vector A u = 3 i n + j n and vector B u = 3 i n - 2j n + 2 k n. What is the cross product A u * B u ?

Force F u = -10j n N is exerted on a particle at r u = 15 i n + 5j n2 m. What is the torque on the particle about the origin?

A 1.3 kg ball on the end of a lightweight rod is located at 1x, y2 = 13.0 m, 2.0 m2, where the y-axis is vertical. The other end of the rod is attached to a pivot at 1x, y2 = 10 m, 3.0 m2. What is the torque about the pivot? Write your answer using unit vectors

What are the magnitude and direction of the angular momentum relative to the origin of the 200 g particle in FIGURE EX12.42?

What is the angular momentum vector of the 2.0 kg, 4.0-cmdiameter rotating disk in FIGURE EX12.43?

What is the angular momentum vector of the 500 g rotating bar in FIGURE EX12.44?

How fast, in rpm, would a 5.0 kg, 22-cm-diameter bowling ball have to spin to have an angular momentum of 0.23 kgm2 /s?

A 2.0 kg, 20-cm-diameter turntable rotates at 100 rpm on frictionless bearings. Two 500 g blocks fall from above, hit the turntable simultaneously at opposite ends of a diameter, and stick. What is the turntables angular velocity, in rpm, just after this event?

A 75 g, 6.0-cm-diameter solid spherical top is spun at 1200 rpm on an axle that extends 1.0 cm past the edge of the sphere. The tip of the axle is placed on a support. What is the tops precession frequency in rpm?

A toy gyroscope has a ring of mass M and radius R attached to the axle by lightweight spokes. The end of the axle is distance R from the center of the ring. The gyroscope is spun at angular velocity v, then the end of the axle is placed on a support that allows the gyroscope to precess. a. Find an expression for the precession frequency in terms of M, R, v, and g. b. A 120 g, 8.0-cm-diameter gyroscope is spun at 1000 rpm and allowed to precess. What is the precession period?

A 300 g ball and a 600 g ball are connected by a 40-cm-long massless, rigid rod. The structure rotates about its center of mass at 100 rpm. What is its rotational kinetic energy?

An 800 g steel plate has the shape of the isosceles triangle shown in FIGURE P12.50. What are the x- and y-coordinates of the center of mass? Hint: Divide the triangle into vertical strips of width dx, then relate the mass dm of a strip at position x to the values of x and dx.

Determine the moment of inertia about the axis of the object shown in FIGURE P12.51

What is the moment of inertia of a 2.0 kg, 20-cm-diameter disk for rotation about an axis (a) through the center, and (b) through the edge of the disk?

Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when d = 0 and when d = L/2.

Calculate the moment of inertia of the rectangular plate in FIGURE P12.54 for rotation about a perpendicular axis through the center

a. A disk of mass M and radius R has a hole of radius r centered on the axis. Calculate the moment of inertia of the disk. b. Confirm that your answer agrees with Table 12.2 when r = 0 and when r = R. c. A 4.0-cm-diameter disk with a 3.0-cm-diameter hole rolls down a 50-cm-long, 20 ramp. What is its speed at the bottom? What percent is this of the speed of a particle sliding down a frictionless ramp?

Consider a solid cone of radius R, height H, and mass M. The volume of a cone is 1 3 pHR2 . a. What is the distance from the apex (the point) to the center of mass? b. What is the moment of inertia for rotation about the axis of the cone? Hint: The moment of inertia can be calculated as the sum of the moments of inertia of lots of small pieces

A persons center of mass is easily found by having the person lie on a reaction board. A horizontal, 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman lies on the reaction board with her feet over the pivot. The scale reads 25 kg. What is the distance from the womans feet to her center of mass?

A 3.0-m-long ladder, as shown in Figure 12.35, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.40. What is the minimum angle the ladder can make with the floor without slipping?

In Figure P12.59, an 80 kg construction worker sits down 2.0 m from the end of a 1450 kg steel beam to eat his lunch. What is the tension in the cable

A 40 kg, 5.0-m-long beam is supported by, but not attached to, the two posts in Figure P12.60. A 20 kg boy starts walking along the beam. How close can he get to the right end of the beam without it falling over?

Your task in a science contest is to stack four identical uniform bricks, each of length L, so that the top brick is as far to the right as possible without the stack falling over. Is it possible, as Figure P12.61 shows, to stack the bricks such that no part of the top brick is over the table? Answer this question by determining the maximum possible value of d.

A 120-cm-wide sign hangs from a 5.0 kg, 200-cm-long pole. A cable of negligible mass supports the end of the rod as shown in Figure P12.62. What is the maximum mass of the sign if the maximum tension in the cable without breaking is 300 N?

A piece of modern sculpture consists of an 8.0-m-long, 150 kg stainless steel bar passing diametrically through a 50 kg copper sphere. The center of the sphere is 2.0 m from one end of the bar. To be mounted for display, the bar is oriented vertically, with the copper sphere at the lower end, then tilted 35 from vertical and held in place by one horizontal steel cable attached to the bar 2.0 m from the top end. What is the tension in the cable

Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheels energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm. a. A motor spins up the flywheel with a constant torque of 50 Nm. How long does it take the flywheel to reach top speed? b. How much energy is stored in the flywheel? c. The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.0 s. What is the average power delivered to the machine? d. How much torque does the flywheel exert on the machine?

Blocks of mass m1 and m2 are connected by a massless string that passes over the pulley in Figure P12.65. The pulley turns on frictionless bearings. Mass m1 slides on a horizontal, frictionless surface. Mass m2 is released while the blocks are at rest. a. Assume the pulley is massless. Find the acceleration of m1 and the tension in the string. This is a Chapter 7 review problem. b. Suppose the pulley has mass mp and radius R. Find the acceleration of m1 and the tensions in the upper and lower portions of the string. Verify that your answers agree with part a if you set mp = 0.

The 2.0 kg, 30-cm-diameter disk in Figure P12.66 is spinning at 300 rpm. How much friction force must the brake apply to the rim to bring the disk to a halt in 3.0 s?

A 30-cm-diameter, 1.2 kg solid turntable rotates on a 1.2-cm-diameter, 450 g shaft at a constant 33 rpm. When you hit the stop switch, a brake pad presses against the shaft and brings the turntable to a halt in 15 seconds. How much friction force does the brake pad apply to the shaft?

Your engineering team has been assigned the task of measuring the properties of a new jet-engine turbine. Youve previously determined that the turbines moment of inertia is 2.6 kgm2 . The next job is to measure the frictional torque of the bearings. Your plan is to run the turbine up to a predetermined rotation speed, cut the power, and time how long it takes the turbine to reduce its rotation speed by 50,. Your data are given in the table. Draw an appropriate graph of the data and, from the slope of the best-fit line, determine the frictional torque.

A hollow sphere is rolling along a horizontal floor at 5.0 m/s when it comes to a 30 incline. How far up the incline does it roll before reversing directio

A 750 g disk and a 760 g ring, both 15 cm in diameter, are rolling along a horizontal surface at 1.5 m/s when they encounter a 15 slope. How far up the slope does each travel before rolling back down?

A cylinder of radius R, length L, and mass M is released from rest on a slope inclined at angle u. It is oriented to roll straight down the slope. If the slope were frictionless, the cylinder would slide down the slope without rotating. What minimum coefficient of static friction is needed for the cylinder to roll down without slipping?

The 5.0 kg, 60-cm-diameter disk in FIGURE P12.72 rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released. a. What is the cylinders initial angular acceleration? b. What is the cylinders angular velocity when it is directly below the axle?

A thin, uniform rod of length L and mass M is placed vertically on a horizontal table. If tilted ever so slightly, the rod will fall over. a. What is the speed of the center of mass just as the rod hits the table if theres so much friction that the bottom tip of the rod does not slide? b. What is the speed of the center of mass just as the rod hits the table if the table is frictionless?

A long, thin rod of mass M and length L is standing straight up on a table. Its lower end rotates on a frictionless pivot. A very slight push causes the rod to fall over. As it hits the table, what are (a) the angular velocity and (b) the speed of the tip of the rod?

The marble rolls down the track shown in FIGURE P12.75 and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

The sphere of mass M and radius R in FIGURE P12.76 is rigidly attached to a thin rod of radius r that passes through the sphere at distance from the center. A string wrapped around the rod pulls with tension T. Find an expression for the spheres angular acceleration. The rods moment of inertia is negligible. FIGURE P12.76

A satellite follows the elliptical orbit shown in FIGURE P12.77. The only force on the satellite is the gravitational attraction of the planet. The satellites speed at point a is 8000 m/s. a. Does the satellite experience any torque about the center of the planet? Explain. b. What is the satellites speed at point b? c. What is the satellites speed at point c?

A 10 g bullet traveling at 400 m/s strikes a 10 kg, 1.0-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door just after impact?

A 200 g, 40-cm-diameter turntable rotates on frictionless bearings at 60 rpm. A 20 g block sits at the center of the turntable. A compressed spring shoots the block radially outward along a frictionless groove in the surface of the turntable. What is the turntables rotation angular velocity when the block reaches the outer edge?

Luc, who is 1.80 m tall and weighs 950 N, is standing at the center of a playground merry-go-round with his arms extended, holding a 4.0 kg dumbbell in each hand. The merry-go-round can be modeled as a 4.0-m-diameter disk with a weight of 1500 N. Lucs body can be modeled as a uniform 40-cm-diameter cylinder with massless arms extending to hands that are 85 cm from his center. The merry-go-round is coasting at a steady 35 rpm when Luc brings his hands in to his chest. Afterward, what is the angular velocity, in rpm, of the merry-go-round?

A merry-go-round is a common piece of playground equipment. A 3.0-m-diameter merry-go-round with a mass of 250 kg is spinning at 20 rpm. John runs tangent to the merry-go-round at 5.0 m/s, in the same direction that it is turning, and jumps onto the outer edge. Johns mass is 30 kg. What is the merry-gorounds angular velocity, in rpm, after John jumps on?

A 45 kg figure skater is spinning on the toes of her skates at 1.0 rev/s. Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 kg, 20 cm average diameter, 160 cm tall) plus two rod-like arms (2.5 kg each, 66 cm long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 kg, 20-cm-diameter, 200-cm-tall cylinder. What is her new angular velocity, in rev/s?

During most of its lifetime, a star maintains an equilibrium size in which the inward force of gravity on each atom is balanced by an outward pressure force due to the heat of the nuclear reactions in the core. But after all the hydrogen fuel is consumed by nuclear fusion, the pressure force drops and the star undergoes a gravitational collapse until it becomes a neutron star. In a neutron star, the electrons and protons of the atoms are squeezed together by gravity until they fuse into neutrons. Neutron stars spin very rapidly and emit intense pulses of radio and light waves, one pulse per rotation. These pulsing stars were discovered in the 1960s and are called pulsars. a. A star with the mass 1M = 2.0 * 1030 kg2 and size 1R = 7.0 * 108 m2 of our sun rotates once every 30 days. After undergoing gravitational collapse, the star forms a pulsar that is observed by astronomers to emit radio pulses every 0.10 s. By treating the neutron star as a solid sphere, deduce its radius. b. What is the speed of a point on the equator of the neutron star? Your answers will be somewhat too large because a star cannot be accurately modeled as a solid sphere. Even so, you will be able to show that a star, whose mass is 106 larger than the earths, can be compressed by gravitational forces to a size smaller than a typical state in the United States!

The earths rotation axis, which is tilted 23.5 from the plane of the earths orbit, today points to Polaris, the north star. But Polaris has not always been the north star because the earth, like a spinning gyroscope, precesses. That is, a line extending along the earths rotation axis traces out a 23.5 cone as the earth precesses with a period of 26,000 years. This occurs because the earth is not a perfect sphere. It has an equatorial bulge, which allows both the moon and the sun to exert a gravitational torque on the earth. Our expression for the precession frequency of a gyroscope can be written = t/Iv. Although we derived this equation for a specific situation, its a valid result, differing by at most a constant close to 1, for the precession of any rotating object. What is the average gravitational torque on the earth due to the moon and the sun?

The bunchberry flower has the fastest-moving parts ever observed in a plant. Initially, the stamens are held by the petals in a bent position, storing elastic energy like a coiled spring. When the petals release, the tips of the stamen act like medieval catapults, flipping through a 60 angle in just 0.30 ms to launch pollen from anther sacs at their ends. The human eye just sees a burst of pollen; only high-speed photography reveals the details. As FIGURE CP12.85 shows, we can model the stamen tip as a 1.0@mm@long, 10 mg rigid rod with a 10 mg anther sac at the end. Although oversimplifying, well assume a constant angular acceleration. a. How large is the straightening torque? b. What is the speed of the anther sac as it releases its pollen?

The two blocks in FIGURE CP12.86 are connected by a massless rope that passes over a pulley. The pulley is 12 cm in diameter and has a mass of 2.0 kg. As the pulley turns, friction at the axle exerts a torque of magnitude 0.50 Nm. If the blocks are released from rest, how long does it take the 4.0 kg block to reach the floor?

A rod of length L and mass M has a nonuniform mass distribution. The linear mass density (mass per length) is l = cx2 , where x is measured from the center of the rod and c is a constant. a. What are the units of c? b. Find an expression for c in terms of L and M. c. Find an expression in terms of L and M for the moment of inertia of the rod for rotation about an axis through the center.

In FIGURE CP12.88, a 200 g toy car is placed on a narrow 60-cm-diameter track with wheel grooves that keep the car going in a circle. The 1.0 kg track is free to turn on a frictionless, vertical axis. The spokes have negligible mass. After the cars switch is turned on, it soon reaches a steady speed of 0.75 m/s relative to the track. What then is the tracks angular velocity, in rpm?

FIGURE CP12.89 shows a cube of mass m sliding without friction at speed v0. It undergoes a perfectly elastic collision with the bottom tip of a rod of length d and mass M = 2m. The rod is pivoted about a frictionless axle through its center, and initially it hangs straight down and is at rest. What is the cubes velocity both speed and directionafter the collision?

A 75 g, 30-cm-long rod hangs vertically on a frictionless, horizontal axle passing through its center. A 10 g ball of clay traveling horizontally at 2.5 m/s hits and sticks to the very bottom tip of the rod. To what maximum angle, measured from vertical, does the rod (with the attached ball of clay) rotate?