Chapter 7

An old LP record that is originally rotating at 33.3 rad/s is given a uniform angular acceleration of 2.15 rad/s2. Through what angle has the record turned when its angular speed reaches 72.0 rad/s?

948 rad

A scooter has wheels with a diameter of 120 mm. What is the angular speed of the wheels when the scooter is moving forward at 6.00 m/s?

955 rpm

What is the angular speed, in rad/s, of a flywheel turning at 813.0 rpm?

85.14 rad/s

The center of mass (or center of gravity) of a two-particle system is at the origin. One particle is located at (3.0 m, 0.0 m) and has a mass of 2.0 kg. The other particle has a mass of 3.0 kg. What is the location of the 3.0-kg particle?

(-2.0 m, 0.0 m)

Three masses are located in the x-y plane as follows: a mass of 6 kg is at (0 m, 0 m), a mass of 4 kg is at (3 m, 0 m), and a mass of 2 kg is at (0 m, 3 m). Where is the center of mass (or center of gravity) of the system?

(1 m, 0.5 m)

Three masses, 1.0 kg, 2.0 kg, and 3.0 kg, are located at (0.0 m, 0.0 m), (1.0 m, 1.0 m), and (2.0 m, -2.0 m), respectively. What is the location of the center of mass (or center of gravity) of this system?

(1.3 m, -0.67 m)

Three small masses are positioned as follows: 2.0 kg at (0.0 m, 0.0 m), 2.0 kg at (2.0 m, 0.0 m), and 4.0 kg at (2.0 m, 1.0 m). Determine the coordinates of the center of mass (or center of gravity) of this system.

(1.5 m, 0.50 m)

As shown in the figure, a 60-cm length of uniform wire, of mass 60 g, is bent into a right triangle. What are the x and y coordinates of the center of mass (or center of gravity) of this triangle?

(10 cm, 3.0 cm)

In the figure, four point masses are placed as shown. Assume that all the numbers in the figure are accurate to two significant figures. What are the x and y coordinates of the center of mass (or center of gravity) of this arrangement?

(2.3 m, 2.8 m)

A mechanic is examining the wheel of a bicycle to adjust the brake. With the bicycle off the ground, he manually rotates the wheel until it reaches an angular speed of 12.0 rad/s and then allows it to coast to a stop. If the wheel has a moment of inertia of 0.100 kg • m2, and the wheel slows uniformly to a stop in 160 s, what is the magnitude of the retarding torque?

0.00750 N • m

A 2.0-m rope is lying on a table. You pick up one end and start raising it vertically. How high above the table is the center of mass (or center of gravity) of the rope when half of the rope has lifted off the table?

0.25 m

A uniform solid cylinder of mass 10 kg can rotate about a frictionless axle through its center O, as shown in the cross-sectional view in the figure. A rope wrapped around the outer radius R1 = 1.0 m exerts a force of magnitude F1 = 5.0 N to the right. A second rope wrapped around another section of radius R2 = 0.50 m exerts a force of magnitude F2 = 6.0 N downward. What is the angular acceleration of the cylinder?

0.40 rad/s2

A rolling wheel of diameter of 68 cm slows down uniformly from 8.4 m/s to rest over a distance of 115 m. What is the magnitude of its angular acceleration if there was no slipping?

0.90 rad/s2

A 60.0-kg man stands at one end of a 20.0-kg uniform 10.0-m long board. How far from the man is the center of mass (or center of gravity) of the man-board system?

1.25 m

A potter's wheel has the shape of a solid uniform disk of mass 7.0 kg and radius 0.65 m. It spins about an axis perpendicular to the disk at its center. A small 2.1 kg lump of very dense clay is dropped onto the wheel at a distance 0.41 m from the axis. What is the moment of inertia of the system about the axis of spin?

1.8 kg • m2

By what factor will the moment of inertia change?

1/16

Which of the following does not describe its angular position?

1/6 πrad

The drive chain in a bicycle is applying a torque of 0.850 N • m to the wheel of the bicycle. You can treat the wheel as a thin uniform hoop (or ring) with a mass of 0.750 kg and a radius of 33.0 cm. What is the angular acceleration of the wheel?

10.4 rad/s2

A wheel rotates through an angle of 13.8 rad as it slows down uniformly from 22.0 rad/s to 13.5 rad/s. What is the magnitude of the angular acceleration of the wheel?

10.9 rad/s2

A string is wound tightly around a fixed pulley having a radius of 5.0 cm. As the string is pulled, the pulley rotates without any slipping of the string. What is the angular speed of the pulley when the string is moving at 5.0 m/s?

100 rad/s

A pulley has an initial angular speed of 12.5 rad/s and a constant angular acceleration of 3.41 rad/s2. Through what angle does the pulley turn in 5.26 s?

113 rad

A triatomic molecule is oriented as follows along the x-axis: mass m is at the origin, mass 2m is at x = a, and, mass 3m is at x = 2a. What is the moment of inertia of this molecule about the y-axis?

14ma2

What is the approximate speed of the skater's hand, which is located about 0.8 m from the rotation axis?

15 m/s

What is its moment of inertia about an axis through its center, perpendicular to the disk?

2.7×10−5kg⋅m2

As shown in the figure, a 35.30-kg box is attached to a light string that is wrapped around a cylindrical frictionless spool of radius 10.0 cm and moment of inertia 4.00 kg • m2. The spool is suspended from the ceiling, and the box is then released from rest a distance 3.50 m above the floor. How long does it take for the box to reach the floor?

2.97 s

A 30-kg child stands at one end of a floating 20-kg canoe that is 5.0-m long and initially at rest in the water. The child then slowly walks to the other end of the canoe. How far does the canoe move in the water, assuming water friction is negligible?

3.0 m

Part complete Assume that ϕ = 60 ∘ and a = 1.0 mm . How large is the "straightening torque"? (You can omit gravitational forces from your calculation; the gravitational torque is much less than this.)

3.1×10−7 N⋅m

A wheel accelerates from rest to 59 rad/s at a uniform rate of 58 rad/s2. Through what angle (in radians) did the wheel turn while accelerating?

30 rad

The lug nuts on a car wheel require tightening to a torque of 90 N • m. If a 30-cm long wrench is used, what is the magnitude of the minimum force required using the wrench?

300 N

What is the approximate centripetal acceleration of the skater's hand?

300 m/s2

A 95-N force exerted at the end of a 0.32-m long torque wrench produces a torque of 15 N • m. What is the angle (less than 90°) between the wrench handle and the direction of the applied force?

30°

A chicken is running in a circular path with an angular speed of 1.52 rad/s. How long does it take the chicken to complete one revolution?

4.13 s

A man in a gym is holding an 8.0-kg weight at arm's length, a distance of 0.55 m from his shoulder joint. What is the torque about his shoulder joint due to the weight if his arm is horizontal?

43 N • m

A uniform solid cylinder of mass 10 kg can rotate about a frictionless axle through its center O, as shown in the cross-sectional view in the figure. A rope wrapped around the outer radius R1 = 1.0 m exerts a force of magnitude F1 = 5.0 N to the right. A second rope wrapped around another section of radius R2 = 0.50 m exerts a force of magnitude F2 = 6.0 N downward. How many radians does the cylinder rotate through in the first 5.0 seconds, if it starts from rest?

5.0 rad

How long does it take for a rotating object to speed up from 15.0 rad/s to 33.3 rad/s if it has a uniform angular acceleration of 3.45 rad/s2?

5.30 s

A 385-g tile hangs from one end of a string that goes over a pulley with a moment of inertia of 0.0125 kg • m2 and a radius of 15.0 cm. A mass of 710 g hangs from the other end of the string. When the tiles are released, the larger one accelerates downward while the lighter one accelerates upward. The pulley has no friction in its axle and turns without the string slipping. What is the tension in the string on the side of the 710-g tile?

5.59 N

The drum shown in the figure has a radius of 0.40 m and a moment of inertia of 2.3 kg • m2about an axis perpendicular to it through its center. The frictional torque at the drum axle is 3.0 N • m. A 14-m length of rope is wound around the rim. The drum is initially at rest. A constant force is applied to the free end of the rope until the rope is completely unwound and slips off. At that instant, the angular speed of the drum is 23 rad/s. The drum then slows down at a constant rate and comes to a halt. In this situation, what was the magnitude of the constant force applied to the rope?

51 N

A machinist turns on the power on to a grinding wheel at time t = 0 s. The wheel accelerates uniformly from rest for 10 s and reaches the operating angular speed of 58 rad/s. The wheel is run at that angular velocity for 30 s, and then power is shut off. The wheel slows down uniformly at 1.4 rad/s2 until the wheel stops. What is the total number of revolutions made by the wheel in this situation?

510

A Ferris wheel rotating at 20 rad/s slows down with a constant angular acceleration of magnitude 5.0 rad/s2. How many revolutions does it make while slowing down before coming to rest?

6.4

A torque of 12 N • m is applied to a solid, uniform disk of radius 0.50 m. If the disk accelerates at 1.6 rad/s2, what is the mass of the disk?

60 kg

A machinist turns on the power on to a grinding wheel at time t = 0 s. The wheel accelerates uniformly from rest for 10 s and reaches the operating angular speed of 96 rad/s. The wheel is run at that angular velocity for 40 s and then power is shut off. The wheel slows down uniformly at 1.5 rad/s2 until the wheel stops. For how long a time after the power is shut off does it take the wheel to stop?

64 s

What force should you apply?

68 N

A uniform rod is 2.0 m long. It is hinged to a wall at its left end, and held in a horizontal position at its right end by a vertical very light string, as shown in the figure. What is the angular acceleration of the rod at the moment after the string is released if there is no friction in the hinge?

7.4 rad/s2

A force of 17 N is applied to the end of a 0.63-m long torque wrench at an angle 45° from a line joining the pivot point to the handle. What is the magnitude of the torque about the pivot point produced by this force?

7.6 N • m

Which force shown in the figure will apply the greatest torque to the nut?

C

What should you do if you want to hold a baseball bat on your palm so that it balances vertically?

Put the end of the handle in your palm, with the barrel up.

What is the gravitational torque on your arm in this position, computed around the shoulder joint?

τ=15N⋅m