Answer: An object rotates about a fixed axis, and the

Figure 10-19 is a graph of the angular velocity versus time for a disk rotating like a merry-go-round. For a point on the disk rim, rank the instants a, b, c, and d according to the magnitude of the (a) tangential and (b) radial acceleration, greatest first.

Figure 10-20 shows plots of angular position B versus time t for three cases in which a disk is rotated like a merry-go-round. In each case, the rotation direction changes at a certain angular position Beilange' (a) For each case, determine whether 8 Behange is clockwise or counterclock- _9001----'----'----) -, wise from B = 0, or whether it is at B = O. For each case, determine Fig. 10-20 Question 2. (b) whether OJ is zero before, after, or at t = 0 and (c) whether a' is positive, negative, or zero.

A force is applied to the rim of a disk that can rotate like a merry-go-round, so as to change its angular velocity. Its initial and final angular velocities, respectively, for four situations are: (a) -2 rad/s, 5 rad/s; (b) 2 rad/s, 5 rad/s; ( c) - 2 rad/s, - 5 rad/s; and (d) 2 rad/s, -5 rad/s. Rank the situations according to the work done by the torque due to the force, greatest first.

Figure 10-21b is a graph of the angular position of the rotating disk of Fig. 1O-21a. Is the angular velocity of the disk positive, negative, or zero at (a) t = 1 s, (b) t = 2 s, and (c) t = 3 s? (d) Is the angular acceleration positive or negative?

In Fig. 10-22, two forces FI and fz act on a disk that turns about its center like a merry-go-round. The forces maintain the indicated angles during the rotation, which is counterclockwise and at a constant rate. However, we are to decrease the angle B of FI without changing the magnitude of Fl' (a) To keep the angular speed constant, should we increase, decrease, or maintain the magnitude of Fig. 10-22 Question 5. fz? Do forces (b) FI and (c) fz tend to rotate the disk clockwise or counterclockwise?

In the overhead view of Fig. 10-23, five forces of the same magnitude act on a strange merry-go-round; it is a square that can rotate about point P, at rnidlength along one of the edges. Rank the forces according to the magnitude of the torque they create about point P, greatest first.

Figure 1O-24a is an overhead view of a horizontal bar that can pivot; two horizontal forces act on the bar, but it is stationary. If the angle between the bar and fz is now decreased from 90 and the bar is still not to turn, should F2 be made larger, made smaller, or left the same?

Figure 1O-24b shows an overhead view of a horizontal bar that is rotated about the pivot point by two horizontal forces, FI and fz, with fz at angle if; to the bar. Rank the following values of if; according to the magnitude of the angular acceleration of the bar, greatest first: 90,70, and 110.

Figure 10-25 shows a uniform metal plate that had been square before 25% of it was snipped off. Three lettered points are indicated. Rank them according to the rotational inertia of the plate around a perpendicular axis through them, greatest first.

Figure 10-26 shows three flat disks (of the same a[jj7: lc ........ ...... b .. ;;.~ I iJ __ ~c Fig. 10-25 Question 9. radius) that can rotate about their centers like merry-go-rounds. Each disk consists of the same two materials, one denser than the other (density is mass per unit volume). In disks 1 and 3, the denser material forms the outer half of the disk area. In disk 2, it forms the inner half of the disk area. Forces with identical magnitudes are applied tangentially to the disk, either at the outer edge or at the interface of the two materials, as shown. Rank the disks according to (a) the torque about the disk center, (b) the rotational inertia about the disk center, and (c) the angular acceleration of the disk, greatest first.

A disk, initially rotating at 120 rad/s, is slowed down with a constant angular acceleration of magnitude 4.0 rad/s2. (a) How much time does the disk take to stop? (b) Through what angle does the disk rotate during that time?

The angular speed of an automobile engine is increased at a constant rate from 1200 rev/min to 3000 rev/min in 12 s. (a) What is its angular acceleration in revolutions per minute-squared? (b) How many revolutions does the engine make during this 12 s interval?

A flywheel turns through 40 rev as it slows from an angular speed of 1.5 rad/s to a stop. (a) Assuming a constant angular acceleration, find the time for it to come to rest. (b) What is its angular acceleration? (c) How much time is required for it to complete the first 20 of the 40 revolutions?

A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at 10 rev/s; 60 revolutions later, its angular speed is 15 rev/so Calculate (a) the angular acceleration, (b) the time required to complete the 60 revolutions, (c) the time required to reach the 10 rev/s angular speed, and (d) the number of revolutions from rest until the time the disk reaches the 10 rev/s angular speed.

A wheel has a constant angular acceleration of 3.0 rad/s2. During a certain 4.0 s interval, it turns through an angle of 120 rad. Assuming that the wheel started from rest, how long has it been in motion at the start of this 4.0 s interval?

A merry-go-round rotates from rest with an angular acceleration of 1.50 rad/s2. How long does it take to rotate through (a) the first 2.00 rev and (b) the next 2.00 rev?

At t = 0, a flywheel has an angular velocity of 4.7 rad/s, a constant angular acceleration of -0.25 rad/s2, and a reference line at 00 = O. (a) Through what maximum angle Omax will the reference line turn in the positive direction? What are the (b) first and (c) second times the reference line will be at 0 = ~Omax? At what (d) negative time and (e) positive time will the reference line be at o = 10.5 rad? (f) Graph 0 versus (, and indicate the answers to (a) through (e) on the graph.

If an airplane propeller rotates at 2000 rev/min while the airplane flies at a speed of 480 km/h relative to the ground, what is the linear speed of a point on the tip of the propeller, at radius 1.5 m, as seen by (a) the pilot and (b) an observer on the ground? The plane'S velocity is parallel to the propeller's axis of rotation.

What are the magnitudes of (a) the angular velocity, (b) the radial acceleration, and (c) the tangential acceleration of a spaceship taking a circular turn of radius 3220 km at a speed of 29 000 kmlh?

An object rotates about a fixed axis, and the angular position of a reference line on the object is given byB = OAOe2t, where Bis in radians and t is in seconds. Consider a point on the object that is 4.0 cm from the axis of rotation. At t = 0, what are the magnitudes of the point's (a) tangential component of acceleration and (b) radial component of acceleration?

Between 1911 and 1990, the top of the leaning bell tower at Pisa, Italy, moved toward the south at an average rate of 1.2 mm/y. The tower is 55 m tall. In radians per second, what is the average angular speed of the tower's top about its base?

An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10 m and, in starting, rotates according to 17 = 0.30t2, where t is in seconds and 17 is in radians. When t = 5.0 s, what are the magnitudes of the astronaut's (a) angular velocity, (b) linear velocity, (c) tangential acceleration, and (d) radial acceleration?

A flywheel with a diameter of 1.20 m is rotating at an angular speed of 200 rev/min. (a) What is the angular speed of the flywheel in radians per second? (b) What is the linear speed of a point on the rim of the flywheel? (c) What constant angular acceleration (in revolutions per minute-squared) will increase the wheel's angular speed to 1000 rev/min in 60.0 s? (d) How many revolutions does the wheel make during that 60.0 s?

A vinyl record is played by rotating the record so that an approximately circular groove in the vinyl slides under a stylus. Bumps in the groove run into the stylus, causing it to oscillate. The equipment converts those oscillations to electrical signals and then to sound. Suppose that a record turns at the rate of 33~ rev/min, the groove being played is at a radius of 10.0 cm, and the bumps in the groove are uniformly separated by 1.75 mm. At what rate (hits per second) do the bumps hit the stylus?

(a) What is the angular speed w about the polar axis of a point on Earth's surface at latitude 40 N? (Earth rotates about that axis.) (b) What is the linear speed v of the point? What are (c) wand (d) v for a point at the equator?

The flywheel of a steam engine runs with a constant angular velocity of 150 rev/min. When steam is shut off, the friction of the bearings and of the air stops the wheel in 2.2 h. (a) What is the constant angular acceleration, in revolutions per minute-squared, of the wheel during the slowdown? (b) How many revolutions does the wheel make before stopping? (c) At the instant the flywheel is turning at 75 rev/min, what is the tangential component of the linear acceleration of a flywheel particle that is 50 cm from the axis of rotation? (d) What is the magnitude of the net linear acceleration of the particle in (c)?

A record turntable is rotating at 33~ rev/min. A watermelon seed is on the turntable 6.0 cm from the axis of rotation. (a) Calculate the acceleration of the seed, assuming that it does not slip. (b) What is the minimum value of the coefficient of static friction between the seed and the turntable if the seed is not to slip? (c) Suppose that the turntable achieves its angular speed by starting from rest and undergoing a constant angular acceleration for 0.25 s. Calculate the minimum coefficient of static friction required for the seed not to slip during the acceleration period.

In Fig. 10-28, wheel A of radius rA = 10 cm is coupled by belt B to wheel C of radius rc = 25 cm. The angular speed of wheel A is increased from rest at a constant rate of 1.6 rad/s2 Find the time needed for wheel C to reach an angular speed of 100 rev/min, assuming the belt does not slip. (Hint: If the belt does not slip, the linear speeds at the two rims must be equal.)

An early method of measuring the speed of light makes use of a rotating slotted wheel. A beam of light passes through one of the slots at the outside edge of the wheel, as in Fig. 10-29 , travels to a distant mirror, and returns to the wheel just in time to pass through the next slot in the wheel. One such slotted wheel has a radius of 5.0 cm and 500 slots around its edge. Measurements taken when the mirror is L = 500 m from the wheel indicate a speed of light of 3.0 X 105 km/s. (a) What is the (constant) angular speed of the wheel? (b) What is the linear speed of a point on the edge of the wheel?

A gyroscope flywheel of radius 2.83 cm is accelerated from rest at 14.2 rad/s2 until its angular speed is 2760 rev/min. (a) What is the tangential acceleration of a point on the rim of the flywheel during this spin-up process? (b) What is the radial acceleration of this point when the flywheel is spinning at full speed? (c) Through what distance does a point on the rim move during the spin-up?

A disk, with a radius of 0.25 m, is to be rotated like a merry-go-round through 800 rad, starting from rest, gaining angular speed at the constant rate aj through the first 400 rad and then losing angular speed at the constant rate -aj until it is again at rest. The magnitude of the centripetal acceleration of any portion of the disk is not to exceed 400 m/s2 (a) What is the least time required for the rotation? (b) What is the corresponding value of aj?

A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period T of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of T = 0.033 s that is increasing at the rate of 1.26 X 10-5 sly. (a) What is the pulsar's angular acceleration a? (b) If a is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year 1054. Assuming constant a, find the initial T.

Calculate the rotational inertia of a wheel that has a kinetic energy of 24 400 J when rotating at 602 rev/min.

Figure 10-30 gives angular speed versus time for a thin rod that rotates around one end. The scale on the w axis is set by Ws = 6.0 rad/s. (a) What is the magnitude of the rod's angular acceleration? (b) At t = 4.0 s, the rod has a rotational kinetic energy of 1.60 J. What is its kinetic energy at t = O?

Two uniform solid cylinders, each rotating about its central (longitudinal) axis at 235 rad/s, have the same mass of 1.25 kg but differ in radius. What is the rotational kinetic energy of (a) the smaller cylinder, of radius 0.25 m, and (b) the larger cylinder, of radius 0.75 m?

Figure 10-31a shows a disk that can rotate about an axis at a radial distance h from the center of the disk. Figure 10-31b gives the rotational inertia I of the disk about the axis as a function of that distance h, from the center out to the edge of the disk. The scale on the I axis is set by IA = 0.050 kg' m2 and IB = 0.150 kg m2 What is the mass of the disk?

Calculate the rotational inertia of a meter stick, with mass 0.56 kg, about an axis perpendicular to the stick and located at the 20 cm mark. (Treat the stick as a thin rod.)

Figure 10-32 shows three 0.0100 kg particles that have been glued to a rod of length L = 6.00 cm and negligible mass. The assembly can rotate around a perpendicular axis through point at the left end. If we remove one particle (that is, 33% of the mass), by what percentage does the rotational inertia of the assembly 11-' --L----II Fig. 1 0-32 Problems 38 and 62. PROBLEMS 269 around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one?

Trucks can be run on energy stored in a rotating flywheel, with an electric motor getting the flywheel up to its top speed of 2001T rad/s. One such flywheel is a solid, uniform cylinder with a mass of 500 kg and a radius of 1.0 m. (a) What is the kinetic energy of the flywheel after charging? (b) If the truck uses an average power of 8.0 kW, for how many minutes can it operate between chargings?

Figure 10-33 shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length L = 1.0000 m and (total) mass M = 100.0 mg. The disk arrangement can rotate about a perpendicular axis through its central disk at point 0. (a) What is the rotational inertia of the arrangement about that axis? (b) If we approximated the arrangement as being a uniform rod of mass M and length L, what percentage error would we make in using the formula in Table 10-2e to calculate the rotational inertia?

In Fig. 10-34, two particles, each with mass m = 0.85 kg, are fastened to each other, and to a rotation axis at 0, by two thin rods, each with length d = 5.6 cm and mass M = 1.2 kg. The combination rotates around the rotation axis with the angular speed w = 0.30 rad/s. Measured about 0, what are the combination's (a) rotational inertia and (b) kinetic energy?

The masses and coordinates of four particles are as follows: 50 g, x = 2.0 cm, Y = 2.0 cm; 25 g, x = 0, Y = 4.0 cm; 25 g, x = - 3.0 cm,Y = -3.0 cm; 30 g,x = -2.0 cm,Y = 4.0 cm. What are the rotational inertias of this collection about the (a) x, (b) Y, and (c) z axes? (d) Suppose the answers to (a) and (b) are A and B, respectively. Rotation Then what is the answer to (c) in axis terms of A and B?

The uniform solid T block in Fig. 10-35 has mass 0.172 kg and edge lengths a = 3.5 cm, b = 8.4 cm, and c = 1.4 cm. Calculate its rotational inertia about an axis through one corner and perpendicular to the large faces.

Four identical particles of mass 0.50 kg each are placed at the vertices of a 2.0 m X 2.0 m square and held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is perpendicular to the plane of the square, and (c) lies in the plane of the square and passes through two diagonally opposite particles?

The body in Fig. 10-36 is pivoted at 0, and two forces act on it as shown. If '1 = 1.30 m, r2 = 2.15 m, Fj = 4.20 N, F2 = 4.90 N, OJ = 75.0, and O2 = 60.0, what is the net torque about the pivot?

The body in Fig. 10-37 is Fig. 10-36 Problem 45. pivoted at O. Three forces act on it: FA = 10 N at point A, 8.0 m from 0; FB = 16 Nat B, 4.0 m from 0; and Fe = 19 Nat C, 3.0 m from O. What is the net torque about O?

A small ball of mass 0.75 kg is attached to one end of a 1.25-m-Iong massless rod, and the other end of the rod is hung from a pivot. When the resulting pendulum is 30 from the vertical, what is the magnitude of the gravitational torque calculated about the pivot?

The length of a bicycle pedal arm is 0.152 m, and a downward force of 111 N is applied to the pedal by the rider. What is the magnitude of the torque about the pedal arm's pivot when the arm is at angle (a) 30, (b) 90, and (c) 180 with the vertical?

During the launch from a board, a diver's angular speed about her center of mass changes from zero to 6.20 rad/s in 220 ms. Her rotational inertia about her center of mass is 12.0 kg m2 During the launch, what are the magnitudes of (a) her average angular acceleration and (b) the average external torque on her from the board?

If a 32.0 N . m torque on a wheel causes angular acceleration 25.0 rad/s2, what is the wheel's rotational inertia?

In Fig. 10-38, block 1 has mass n11 = 460 g, block 2 has mass /112 = 500 g, and the pulley, which is mounted on a horizontal axle with negligible friction, has radius R = 5.00 cm. When released from rest, block 2 falls 75.0 cm in 5.00 s without the cord slipping on the pulley. (a) What is the magnitude of the acceler- IIlj 1112 ation of the blocks? What are (b) ten- Fig. 10-38 sion T2 and (c) tension Ii? (d) What is Problems 51 and 83. the magnitude of the pulley's angular acceleration? (e) What is its rotational inertia?

In Fig. 10-39, a cylinder having a mass of2.0 kg can rotate about its central axis through point O. Forces are applied as shown: Fj = 6.0 N, F2 = 4.0 N, F3 = 2.0 N, and F4 = 5.0 N. Also, r = 5.0 cm and R = 12 cm. Find the (a) magnitude and (b) direction of the angular acceleration of the cylinder. (During the rotation, the forces maintain their same angles relative to the cylinder.)

Figure 10-40 shows a uniform disk that can rotate around its center like a merry-goround. The disk has a radius of 2.00 cm and a mass of 20.0 grams and is initially at rest. Starting at time t = 0, two forces are to be apFig. 1 0-40 Problem 53. plied tangentially to the rim as indicated, so that at time t = 1.25 s the disk has an angular velocity of 250 rad/s counterclockwise. Force ~ has a magnitude of 0.100 N. What is magnitude F2?

In a judo foot-sweep move, you sweep your opponent's left foot out from under him while pulling on his gi (uniform) toward that side. As a result, your opponent rotates around his right foot and onto the mat. Figure 10-41 shows a simplified diagram of your opponent as you face him, with his left foot swept out. The rotational axis is throughyoint O. The gravitational force Fg on him effectively acts at his center of mass, which is a horizontal distance d = 28 cm from point O. His mass is 70 kg, and his rotational inertia about point 0 is 65 kg m2 What is the magnitUde of his initial angular acceleration about point 0 if your pull Fa on his Fig. 10-41 h Problem 54. gi is (a) negligible and (b) horizontal with a magnitude of 300 Nand applied at height h = 1.4 m?

In Fig. 10-42a, an irregularly shaped plastic plate with uniform thickness and density (mass per unit volume) is to be rotated around an axle that is perpendicular to the plate face and through point O. The rotational inertia of the plate about that axle is measured with the following method. A circular disk of mass 0.500 kg and radius 2.00 cm is glued to the plate, with its center aligned with point 0 (Fig. 10-42b). A string is wrapped around the edge of the disk the way a string is wrapped around a top. Then the string is pulled for 5.00 s. As a result, the disk and plate are rotated by a constant force of 00400 N that is applied by the string tangentially to the edge of the disk. The resulting angular speed is 114 rad/s. What is the rotational inertia of the plate about the axle?

Figure 10-43 shows rLl-'+-1'1 particles 1 and 2, each of I==T===========-e mass m, attached to the ends 1 Ai 2 of a rigid massless rod of length Ll + L 2, with Ll = 20 Fig. 1 0-43 Problem 56. cm and L2 = 80 cm. The rod is held horizontally on the fulcrum and then released. What are the magnitudes of the initial accelerations of (a) particle 1 and (b) particle 2?

A pulley, with a rotational inertia of 1.0 X 10-3 kg m2 about its axle and a radius of 10 cm, is acted on by a force applied tangentially at its rim. The force magnitude varies in time as F = 0.50t + 0.30t2, with F in newtons and t in seconds. The pulley is initially at rest. At t = 3.0 s what are its (a) angular acceleration and (b) angular speed?

(a) If R = 12 cm, M = 400 g, and m = 50 g in Fig. 10-18, find the speed of the block after it has descended 50 cm starting from rest. Solve the problem using energy conservation principles. (b) Repeat (a) withR = 5.0 cm

An automobile crankshaft transfers energy from the engine to the axle at the rate of 100 hp (= 74.6 kW) when rotating at a speed of 1800 rev/min. What torque (in newton-meters) does the crankshaft deliver?

A thin rod of length 0.75 m and mass 0042 kg is suspended freely from one end. It is pulled to one side and then allowed to swing like a pendulum, passing through its lowest position with angular speed 4.0 rad/s. Neglecting friction and air resistance, find (a) the rod's kinetic energy at its lowest position and (b) how far above that position the center of mass rises.

A 32.0 kg wheel, essentially a thin hoop with radius 1.20 m, is rotating at 280 rev/min. It must be brought to a stop in 15.0 S. (a) How much work must be done to stop it? (b) What is the required average power?

In Fig. 10-32, three 0.0100 kg particles have been glued to a rod of length L = 6.00 cm and negligible mass and can rotate around a perpendicular axis through point 0 at one end. How much work is required to change the rotational rate (a) from 0 to 20.0 rad/s, (b) from 20.0 rad/s to 40.0 rad/s, and ( c) from 40.0 rad/s to 60.0 rad/s? (d) What is the slope of a plot of the assembly's kinetic energy (in joules) versus the square of its rotation rate (in radianssquared per second-squared)?

A meter stick is held vertically with one end on the floor and is then allowed to fall. Find the speed of the other end just before it hits the floor, assuming that the end on the floor does not slip. (Hint: Consider the stick to be a thin rod and use the conservation of energy principle.)

A uniform cylinder of radius 10 cm and mass 20 kg is mounted so as to rotate freely about a horizontal axis that is parallel to and 5.0 cm from the central longitudinal axis of the cylinder. (a) What is the rotational inertia of the cylinder about the axis of rotation? (b) If the cylinder is released from rest with its central longitudinal axis at the same height as the axis about which the cylinder rotates, what is the angular speed of the cylinder as it passes through its lowest position?

A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length 55.0 m. At the instant it makes an angle of 35.0 with the vertical as it falls, what are (a) the radial acceleration of the top, and (b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.) (c) At what angle 0 is the tangential acceleration equal to g?

A uniform spherical shell of mass M = 4.5 kg and radius R = 8.5 cm can rotate about a vertical axis on frictionless bearings (Fig. 10-44). A massless cord passes around the equator of the shell, over a pulley of rotational inertia 1 = 3.0 X 10-3 kg . m2 and radius r = 5.0 cm, and is attached to a small object of mass m = 0.60 kg. There is no friction on the pulley's axle; the cord does not slip on the pUlley. What is the speed of the object when it has fallen 82 cm after being released from rest? Use energy considerations.

Figure 10-45 shows a rigid assembly of a thin hoop (of mass m and radius R = 0.150 m) and a thin radial rod (of mass m and length L = 2.00R). The assembly is upright, but if we give it a slight nudge, it will rotate around a horizontal axis in the plane of the rod and hoop, through the Hoop Rod Rotation -----axis lower end of the rod. Assuming that Fig. 10-45 Problem 67. the energy given to the assembly in such a nudge is negligible, what would be the assembly's angular speed about the rotation axis when it passes through the upsidedown (inverted) orientation?

Two uniform solid spheres have the same mass of 1.65 kg, but one has a radius of 0.226 m and the other has a radius of 0.854 m. Each can rotate about an axis through its center. (a) What is the magnitude r of the torque required to bring the smaller sphere from rest to an angular speed of 317 rad/s in 15.5 s? (b) What is the magnitude F of the force that must be applied tangentially at the sphere's equator to give that torque? What are the corresponding values of (c) rand (d) Ffor the larger sphere?

In Fig. 10-46, a small disk of radius r = 2.00 cm has been glued to the edge of a larger disk of radius R = 4.00 cm so that the disks lie in the same plane. The disks can be rotated around a perpendicular axis through point 0 at the center of the larger disk. The disks both have a uniform density (mass per unit volume) of 1.40 X 103 kg/m3 and a uniform thickness of 5.00 mm. What is the rotational inertia of the two-disk assembly about the rotation axis through O?

A wheel, starting from rest, rotates with a constant angular acceleration of 2.00 rad/s2. During a certain 3.00 s interval, it turns through 90.0 rad. (a) What is the angular velocity of the wheel at the start of the 3.00 s interval? (b) How long has the wheel been turning before the start of the 3.00 s interval?

In Fig. 10-47, two 6.20 kg blocks are connected by a massless string over a pulley of radius 2.40 cm and rotational inertia 7.40 X 10-4 kg m2 The string does not slip on the pulley; it is not known whether there is friction between the table and the sliding block; the Fig. 10-47 Problem 71. pulley's axis is frictionless. When this system is released from rest, the pulley turns through 0.650 rad in 91.0 ms and the acceleration of the blocks is constant. What are (a) the magnitude of the pulley's angular acceleration, (b) the magnitude of either block's acceleration, (c) string tension TJ, and (d) string tension T2?

Attached to each end of a thin steel rod of length 1.20 m and mass 6.40 kg is a small ball of mass 1.06 kg. The rod is constrained to rotate in a horizontal plane about a vertical axis through its midpoint. At a certain instant, it is rotating at 39.0 rev/so Because of friction, it slows to a stop in 32.0 S. Assuming a constant retarding torque due to friction, compute (a) the angular acceleration, (b) the retarding torque, (c) the total energy transferred from mechanical energy to thermal energy by friction, and (d) the number of revolutions rotated during the 32.0 S. (e) Now suppose that the retarding torque is known not to be constant. If any of the quantities (a), (b), ( c), and (d) can still be computed without additional information, give its value.

A uniform helicopter rotor blade is 7.80 m long, has a mass of 110 kg, and is attached to the rotor axle by a single bolt. (a) What is the magnitude of the force on the bolt from the axle when the rotor is turning at 320 rev/min? (Hint: For this calculation the blade can be considered to be a point mass at its center of mass. Why?) (b) Calculate the torque that must be applied to the rotor to bring it to full speed from rest in 6.70 S. Ignore air resistance. (The blade cannot be considered to be a point mass for this calculation. Why not? Assume the mass distribution of a uniform thin rod.) (c) How much work does the torque do on the blade in order for the blade to reach a speed of 320 rev/min?

Racing disks. Figure 10-48 shows two disks that can rotate about their centers like a merry-goround. At time t = 0, the reference lines of the two disks have the same Disk A DiskB orientation. Disk A is already rotat- Fig. 10-48 Problem 74. ing, with a constant angular velocity of 9.5 rad/s. Disk B has been stationary but now begins to rotate at a constant angular acceleration of 2.2 rad/s2 (a) At what time twill the reference lines of the two disks momentarily have the same angular displacement B? (b) Will that time t be the first time since t = o that the reference lines are momentarily aligned?

A high-wire walker always attempts to keep his center of mass over the wire (or rope). He normally carries a long, heavy pole to help: If he leans, say, to his right (his com moves to the right) and is in danger of rotating around the wire, he moves the pole to his left (its com moves to the left) to slow the rotation and allow himself time to adjust his balance. Assume that the walker has a mass of 70.0 kg and a rotational inertia of 15.0 kg m2 about the wire. What is the magnitUde of his angular acceleration about the wire if his com is 5.0 cm to the right of the wire and (a) he carries no pole and (b) the 14.0 kg pole he carries has its com 10 cm to the left of the wire?

Starting from rest at t = 0, a wheel undergoes a constant angular acceleration. When t = 2.0 s, the angular velocity of the wheel is 5.0 rad/s. The acceleration continues until t = 20 s, when it abruptly ceases. Through what angle does the wheel rotate in the interval t = 0 to t = 40 s?

A record turntable rotating at 33~ rev/min slows down and stops in 30 s after the motor is turned off. (a) Find its (constant) angular acceleration in revolutions per minute-squared. (b) How many revolutions does it make in this time?

A rigid body is made of three identical thin rods, each with length L = 0.600 m, fastened together in the form of a letter H (Fig. 10-49). The body is free to rotate about a horizontal axis that runs Fig. 10-49 Problem 78. along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of the H is horizontal. What is the angular speed of the body when the plane of the H is vertical?

(a) Show that the rotational inertia of a solid cylinder of mass M and radius R about its central axis is equal to the rotational inertia of a thin hoop of mass M and radius RlVz about its central axis. (b) Show that the rotational inertia I of any given body of mass M about any given axis is equal to the rotational inertia of an equivalent hoop about that axis, if the hoop has the same mass M and a radius k given by k=f{;. The radius k of the equivalent hoop is called the radius of gyration of the given body.

A disk rotates at constant angular acceleration, from angular position 81 = 10.0 rad to angular position 82 = 70.0 rad in 6.00 S. Its angular velocity at 82 is 15.0 rad/s. (a) What was its angular velocity at 81? (b) What is the angular acceleration? (c) At what angular position was the disk initially at rest? (d) Graph 8 versus time t and angular speed w versus t for the disk, from the beginning of the motion (let t = 0 then).

The thin uniform rod in Fig. 10-50 has length 2.0 m and can pivot about a horizontal, frictionless pin through one end. It is released from rest at angle 8 = 40 above the horizontal. Use the principle of conservation of energy to determine the angular speed of the rod as it passes through the horizontal position.

George Washington Gale Ferris, Jr., a civil engineering graduate from Rensselaer ~ ... ...... ....... i' . i/< . . ;: e Pin Fig. 10-50 Problem 81. Polytechnic Institute, built the original Ferris wheel for the 1893 World's Columbian Exposition in Chicago. The wheel, an astounding engineering construction at the time, carried 36 wooden cars, each holding up to 60 passengers, around a circle 76 m in diameter. The cars were loaded 6 at a time, and once all 36 cars were full, the wheel made a complete rotation at constant angular speed in about 2 min. Estimate the amount of work that was required of the machinery to rotate the passengers alone.

In Fig. 10-38, two blocks, of mass ml = 400 g and m2 = 600 g, are connected by a massless cord that is wrapped around a uniform disk of mass M = 500 g and radius R = 12.0 cm. The disk can rotate without friction about a fixed horizontal axis through its center; the cord cannot slip on the disk. The system is released from rest. Find (a) the magnitude of the acceleration of the blocks, (b) the tension Tj in the cord at the left, and (c) the tension T2 in the cord at the right.

At 7: 14 A.M. on June 30, 1908, a huge explosion occurred above remote central Siberia, at latitude 61 N and longitude 102 E; the fireball thus created was the brightest flash seen by anyone before nuclear weapons. The Tunguska Event, which according to one chance witness "covered an enormous part of the sky," was probably the explosion of a stony asteroid about 140 m wide. (a) Considering only Earth's rotation, determine how much later the asteroid would have had to arrive to put the explosion above Helsinki at longitude 25 E. This would have obliterated the city. (b) If the asteroid had, instead, been a metallic asteroid, it could have reached Earth's surface. How much later would such an asteroid have had to anive to put the impact in the Atlantic Ocean at longitude 20 W? (The resulting tsunamis would have wiped out coastal civilization on both sides of the Atlantic.)

A golf ball is launched at an angle of 20 to the horizontal, with a speed of 60 m/s and a rotation rate of 90 rad/s. Neglecting air drag, determine the number of revolutions the ball makes by the time it reaches maximum height

Figure 10-51 shows a flat construction of two circular rings that have a common center and are held together by three rods of negligible mass. The construction, which is initially at rest, can rotate around the common center (like a merry-go- Fig. 10-51 round), where another rod of negligible mass lies. Problem 86. The mass, inner radius, and outer radius of the rings are given in the following table. A tangential force of magnitude 12.0 N is applied to the outer edge of the outer ring for 0.300 s. What is the change in the angular speed of the construction during that time interval?

In Fig. 10-52, a wheel of radius 0.20 m is mounted on a frictionless horizontal axle. A massless cord is wrapped around the wheel and attached to a 2.0 kg box that slides on a frictionless surface inclined at angle (J = 20 with the horizontal. The box accelerates down the surface at 2.0 m/s2. What is the rotational inertia of the wheel about the axle?

A thin spherical shell has a radius of 1.90 m. An applied torque of 960 N . m gives the shell an angular acceleration of 6.20 rad/s2 about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?

A bicyclist of mass 70 kg puts all his mass on each downwardmoving pedal as he pedals up a steep road. Take the diameter of the circle in which the pedals rotate to be 0040 m, and determine the magnitude of the maximum torque he exerts about the rotation axis of the pedals.

The flywheel of an engine is rotating at 25.0 rad/s. When the engine is turned off, the flywheel slows at a constant rate and stops in 20.0 s. Calculate (a) the angular acceleration of the flywheel, (b) the angle through which the flywheel rotates in stopping, and ( c) the number of revolutions made by the flywheel in stopping.

In Fig. 10-18a, a wheel of radius 0.20 m is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is 0040 kg m2. A massless cord wrapped around the wheel's circumference is attached to a 6.0 kg box. The system is released from rest. When the box has a kinetic energy of 6.0 J, what are (a) the wheel's rotational kinetic energy and (b) the distance the box has fallen?

Our Sun is 2.3 X 104 ly (light-years) from the center of our Milky Way galaxy and is moving in a circle around that center at a speed of 250 krn/s. (a) How long does it take the Sun to make one revolution about the galactic center? (b) How many revolutions has the Sun completed since it was formed about 4.5 X 109 years ago?

A wheel of radius 0.20 m is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is 0.050 kg . m2. A massless cord wrapped around Fig. 10-53 Problem 93. the wheel is attached to a 2.0 kg block that slides on a horizontal frictionless surface. If a horizontal force of magnitude P = 3.0 N is applied to the block as shown in Fig. 10-53, what is the magnitude of the angular acceleration of the wheel? Assume the cord does not slip on the wheel.

A car starts from rest and moves around a circular track of radius 30.0 m. Its speed increases at the constant rate of 0.500 m/s2. (a) What is the magnitude of its net linear acceleration 15.0 s later? (b) What angle does this net acceleration vector make with the car's velocity at this time?

The rigid body shown in Fig. 10-54 consists of three particles connected by massless rods. It is to be rotated about an axis perpendicular to its plane through point P. If M = 0040 kg, a = 30 cm, and b = 50 cm, how much work is required to take the body from rest to an angular speed of 5.0 rad/s?

Beverage engineering. The pull Fig. 10-54 Problem 95. tab was a major advance in the engineering design of beverage containers. The tab pivots on a central bolt in the can's top. When you pull upward on one end of the tab, the other end presses downward on a portion of the can's top that has been scored. If you pull upward with a 10 N force, approximately what is the magnitude of the force applied to the scored section? (You will need to examine a can with a pull tab.)

Figure 10-55 shows a propeller blade that rotates at 2000 rev/min about a perpendicular axis at point B. Point A is at the outer tip of the blade, at radial distance 1.50 m. (a) What is the difference in the magnitudes a of the centripetal acceleraFig. 10-55 Problem 97. tion of point A and of a point at radial distance 0.150 m? (b) Find the slope of a plot of a versus radial distance along the blade.

A yo-yo-shaped device mounted on a horizontal frictionless axis is used to lift a 30 kg box as shown in Fig. 10-56. The outer radius R of the device is 0.50 m, and the radius r of the hub is 0.20 m. When a constant horizontal force Papp of magnitude 140 N is applied to a rope wrapped around the outside of the device, the box, which is suspended from a rope wrapped around the hub, has an upward acceleration of magnitude 0.80 mfs2. What is 'L/- ~"lU mount Fig. 10-56 Problem 98. the rotational inertia of the device about its axis of rotation?

A small ball with mass 1.30 kg is mounted on one end of a rod 0.780 m long and of negligible mass. The system rotates in a horizontal circle about the other end of the rod at 5010 rev/min. (a) Calculate the rotational inertia of the system about the axis of rotation. (b) There is an air drag of 2.30 X 10-2 N on the ball, directed opposite its motion. What torque must be applied to the system to keep it rotating at constant speed?

Tho thin rods (each of mass 0.20 kg) are joined together to form a rigid body as shown in Fig. 10-57. One of the rods has length L1 = 0040 m, and the other has length L2 = 0.50 m. What is the rotational inertia of this rigid body about (a) an axis that is perpendicular to the plane of the paper and passes through the center of the shorter 1_ 1 __1_ 1 __ I ,----2 L1 -------r- -2 L1----, rod and (b) an axis that is perpendicular to the plane of the paper Fig. 10-57 Problem 100. and passes through the center of the longer rod?

In Fig. 10-58, four pulleys are connected by two belts. Pulley A (radius 15 cm) is the drive pulley, and it rotates at 10 rad/s. Pulley B (radius 10 cm) is connected by belt 1 to pulley A. Pulley B' (radius 5 cm) is concentric with pulley B and is rigidly attached to it. Pulley C (radius 25 cm) is connected by belt 2 to pulley B'. Calculate (a) the linear speed of a point on belt 1, (b) the an gular speed of pulley B, (c) the angular speed of pulley B', (d) the linear speed of a point on belt 2, and (e) the angular speed of pulley C. (Hint: If the belt between two pulleys does not slip, the linear speeds at the rims of the two pulleys must be equal.)

The rigid object shown in Fig. 10-59 consists of three balls and three connecting rods, with M = 1.6 kg, L = 0.60 m, and () = 30. The balls may be treated as particles, and the connecting rods have negligible mass. Determine the rotational kinetic energy of the object if it has an angular speed of 1.2 rad/s about (a) an axis that passes through point P and is perpendicular to the plane of the figure and (b) an axis that passes through point P, is perpendicular to the rod of length 2L, and lies in the plane of the figure.

In Fig. 10-60, a thin uniform rod (mass 3.0 kg, length 4.0 m) rotates freely about a horizontal axis A that is perpendicular to the rod and passes through a point at distance d = 1.0 m from the end of the rod. The kinetic energy of the rod as it passes through the vertical position is 20 J. (a) What is the rotational inertia of the rod about axis A ? (b) What is the (linear) speed of the end B of the rod as the rod passes through the vertical position? (c) At what angle B will the rod momentarily stop in its upward swing?

Four particles, each of mass, 0.20 kg, are placed at the vertices of a square with sides of length 0.50 m. The particles are connected by rods of negligible mass. This rigid body can rotate in a vertical plane about a horizontal axis A that passes through one of the particles. The body is released from rest with rod AB horizontal (Fig. 10-61). (a) What is the rotational inertia of the body Fig. 10-61 Problem 104. about axis A? (b) What is the angular speed of the body about axis A when rod AB swings through the vertical position?