?A disk rotates at constant angular acceleration, from angular position \(\theta_{1}\)=

Figure 10-20 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 10-21 shows plots of angular position \(\theta\)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\(\theta_{\text {change }}\). (a) For each case, determine whether \(\theta_{\text {change }}\) is clockwise or counterclockwise from \(\theta\)= 0, or whether it is at \(\theta\)= 0. For each case, determine (b) whether \(\boldsymbol{\omega}\) is zero before, after, or at t = 0 and (c) whether ? is positive, negative, or zero. Text Transcription: theta Theta_change omega

Figure 10-22b is a graph of the angular position of the rotating disk of Fig. 10-22a. 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. 10-23, two forces \(\vec{F}_{1}\)1 and \(\vec{F}_{2}\)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 \(\theta\)of \(\vec{F}_{1}\) without changing the magnitude of\(\vec{F}_{1}\). (a) To keep the angular speed constant should we increase,decrease, or maintain the magnitude of \(\vec{F}_{2}\) Do forces (b) \(\vec{F}_{1}\) and (c) \(\vec{F}_{2}\)tend to rotate the disk clockwise or counterclockwise? Text Transcription: Arrow right over F_1 Arrow right over F_2 theta

In the overhead view of Fig. 10-24, five forces of the same magnitude act on a strange merry-go-round; it is a square that can rotate about point P, at midlength along one of the edges. Rank the forces according to the magnitude of the torque they create about point P, greatest first.

Figure 10-25a 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 \(\vec{F}_{2}\)is now decreased from \(90^{\circ}\)and the bar is still not to turn, should \(\vec{F}_{2}\) be made larger, made smaller, or left the same? Text Transcription: arrow right over F_2 90^circ

Figure 10-25b shows an overhead view of a horizontal bar that is rotated about the pivot point by two horizontal forces, \(\overrightarrow{\boldsymbol{F}}_{1}\) and\(\overrightarrow{\boldsymbol{F}}_{2}\), with \(\overrightarrow{\boldsymbol{F}}_{2}\)at angle \(\phi\) to the bar. Rank the following values of \(\phi\) according to the magnitude of the angular acceleration of the bar, greatest first: 9\(90^{\circ}\),\(70^{\circ}\), and\(90^{\circ}\).\ Text Transcription: arrow right over F_1 arrow right over F_2 pi 90^circ 70^circ 110^circ

Figure 10-26 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 10-27 shows three flat disks (of the same 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.

Figure 10-28a shows a meter stick, half wood and half steel, that is pivoted at the wood end at O. A force \(\vec{F}\)is applied to the steel end at a. In Fig. 10-28b, the stick is reversed and pivoted at the steel end at\(O^{\prime}\), and the same force is applied at the wood end at\(a^{\prime}\). Is the resulting angular acceleration of Fig. 10-28a greater than, less than, or the same as that of Fig. 10-28b? Text Transcription: arrow right over F O’ a’

Figure 10-29 shows three disks, each with a uniform distribution of mass. The radii R and masses M are indicated. Each disk can rotate around its central axis (perpendicular to the disk face and through the center). Rank the disks according to their rotational inertias calculated about their central axes, greatest first.

A wheel, starting from rest, rotates with a constant angular acceleration of 2.00 \(\mathrm{rad} / \mathrm{s}^{2}\). 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? Text Transcription: rad/s^2

In Fig. 10-50, two 6.20 kg blocks are connected by a massless string over a pulley of radius 2.40 cm and rotational inertia 7.40 × 10?4 kg · \(\mathbf{m}^{2}\). The string does not slip on the pulley; it is not known whether there is friction between the table and the sliding block; the pulley’s axis is frictionless. When this system is released from rest, the pulley turns through 0.130 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\(T_{1}\), and (d) string tension\(T_{2}\)? Text Transcription: m^2 T_1 T_2

Racing disks. Figure 10-51 shows two disks that can rotate about their centers like a merry-go-round. At time t = 0, the reference lines of the two disks have the same orientation. Disk A is already rotating, 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 \(\mathrm{rad} / \mathrm{s}^{2}\) . (a) At what time t will the reference lines of the two disks momentarily have the same angular displacement\(\theta)? (b) Will that time t be the first time since t = 0 that the reference lines are momentarily aligned? Text Transcription: rad/s^2 theta

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 · \(\mathrm{m}^{2}\) 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? Text Transcription: m^2

A record turntable rotating at \(33 \frac{1}{3}\) 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? Text Transcription: 33 1/3

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. 10-52). The body is free to rotate about a horizontal axis that runs 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 \(R / \sqrt{2}\) 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 \(k=\sqrt{\frac{I}{M}}\). The radius k of the equivalent hoop is called the radius of gyration of the given body. Text Transcription: R/sqrt 2 k= sqrt I /M

A disk rotates at constant angular acceleration, from angular position \(\theta_{1}\)= 10.0 rad to angular position \(\theta_{2}\)= 70.0 rad in 6.00 s. Its angular velocity at \(\theta_{2}\) is 15.0 rad/s. (a) What was its angular velocity at \(\theta_{1})? (b) What is the angular acceleration? (c) At what angular position was the disk initially at rest? (d) Graph \(\theta)versus time t and angular speed \({\omega}\) versus t for the disk, from the beginning of the motion (let t = 0 then). Text Transcription: theta_1 theta_2 theta omega

The thin uniform rod in Fig. 10-53 has length 2.0 m and can pivot about a horizontal, frictionless pin through one end. It is released from rest at angle \(\theta=40^{\circ}\) 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. Text Transcription: theta=40^circ

In Fig. 10-41, two blocks, of mass m1 = 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 \(T_{1}\) in the cord at the left, and (c) the tension \(T_{2}\)in the cord at the right. Text Transcription: T_1 T_2

At 7:14 a.m. on June 30, 1908, a huge explosion occurred above remote central Siberia, at latitude \(61^{\circ}\) N and longitude \(102^{\circ}\)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^{\circ})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 arrive to put the impact in the Atlantic Ocean at longitude \(20^{\circ}\)W? (The resulting tsunamis would have wiped out coastal civilization on both sides of the Atlantic.) Text Transcription: 102^circ 61^circ 25^circ 20^cir

A golf ball is launched at an angle of \(20^{\circ}\) 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. Text Transcription: 20^circ

Figure 10-54 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 merrygo-round), where another rod of negligible mass lies. 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 the time interval?

In Fig. 10-55, 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 \(\theta=20^{\circ})with the horizontal. The box accelerates down the surface at 2.0 \(\mathrm{m} / \mathrm{s}^{2}\). What is the rotational inertia of the wheel about the axle? Text Transcription: theta=20^circ m/s^2

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 \(\mathrm{rad} / \mathrm{s}^{2}\) 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? Text Transcription: rads/^2

In Fig. 10-19a, a wheel of radius 0.20 m is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is 0.40 kg · \(\mathrm{m}^{2}\). 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? Text Transcription: m^2

Our Sun is 2.3 × \(10^{4}\)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 km/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 × \(10^{9}\) years ago? Text Transcription: 10^4 10^9

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 · \(\mathrm{m}^{2}\). A massless cord wrapped around 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. 10-56, what is the magnitude of the angular acceleration of the wheel? Assume the cord does not slip on the wheel. Text Transcription: m^2

The rigid body shown in Fig. 10-57 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 = 0.40 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?

Figure 10-58 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 acceleration 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. 10-59. 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 \(\vec{F}_{\text {app }}\) 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 \(\mathrm{m} / \mathrm{s}^{2}\) . What is the rotational inertia of the device about its axis of rotation? Text Transcription: arrow right over F_app m/s^2

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 × \(10^{-2}\)N on the ball, directed opposite its motion. What torque must be applied to the system to keep it rotating at constant speed? Text Transcription: 10^-2

Two thin rods (each of mass 0.20 kg) are joined together to form a rigid body as shown in Fig. 10-60. One of the rods has length \(L_{1}\) = 0.40 m, and the other has length \(L_{2}\)= 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 rod and (b) an axis that is perpendicular to the plane of the paper and passes through the center of the longer rod? Text Transcription: L_1 L_2

In Fig. 10-61, 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 \(\boldsymbol{B}^{\prime}\) (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\(\boldsymbol{B}^{\prime}\). Calculate (a) the linear speed of a point on belt 1, (b) the angular speed of pulley B, (c) the angular speed of pulley\(\boldsymbol{B}^{\prime}\), (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.) Text Transcription: B’

The rigid object shown in Fig. 10-62 consists of three balls and three connecting rods, with M = 1.6 kg, L = 0.60 m, and\(\theta=30^{\circ}). 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. Text Transcription: theta=30^circ

In Fig. 10-63, 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 \(\theta\)will the rod momentarily stop in its upward swing? Text Transcription: theta

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. 10-64). (a) What is the rotational inertia of the body about axis A? (b) What is the angular speed of the body about axis A when rod AB swings through the vertical position?

A pulley wheel that is 8.0 cm in diameter has a 5.6-m-long cord wrapped around its periphery. Starting from rest, the wheel is given a constant angular acceleration of 1.5 \(\mathrm{rad} / \mathrm{s}^{2}\). (a) Through what angle must the wheel turn for the cord to unwind completely? (b) How long will this take? Text Transcription: rad/s^2

A vinyl record on a turntable rotates at \(33 \frac{1}{3}\) rev/min. (a) What is its angular speed in radians per second? What is the linear speed of a point on the record (b) 15 cm and (c) 7.4 cm from the turntable axis? Text Transcription: 33 1/3

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.

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/s. 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?

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?

George Washington Gale Ferris, Jr., a civil engineering graduate from Rensselaer 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.

A bicyclist of mass 70 kg puts all his mass on each downward-moving pedal as he pedals up a steep road. Take the diameter of the circle in which the pedals rotate to be 0.40 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.

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.

Beverage engineering. The pull 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, what force magnitude acts on the scored section? (You will need to examine a can with a pull tab.)

Cheetahs running at top speed have been reported at an astounding 114 km/h (about 71 mi/h) by observers driving alongside the animals. Imagine trying to measure a cheetah’s speed by keeping your vehicle abreast of the animal while also glancing at your speedometer, which is registering 114 km/h. You keep the vehicle a constant 8.0 m from the cheetah, but the noise of the vehicle causes the cheetah to continuously veer away from you along a circular path of radius 92 m. Thus, you travel along a circular path of radius 100 m. (a) What is the angular speed of you and the cheetah around the circular paths? (b) What is the linear speed of the cheetah along its path? (If you did not account for the circular motion, you would conclude erroneously that the cheetah’s speed is 114 km/h, and that type of error was apparently made in the published reports.)

A point on the rim of a 0.75-m-diameter grinding wheel changes speed at a constant rate from 12 m/s to 25 m/s in 6.2 s. What is the average angular acceleration of the wheel?