Deck 12: Rotation of a Rigid Body

Question 1

Consider a uniform solid sphere of radius R and mass M rolling without slipping. Which form of its kinetic energy is larger, translational or rotational?

A) Its translational kinetic energy is larger than its rotational kinetic energy.
B) Its rotational kinetic energy is larger than its translational kinetic energy.
C) Both forms of energy are equal.
D) You need to know the speed of the sphere to tell.

Answer 1

Its translational kinetic energy is larger than its rotational kinetic energy.

Question 2

If an irregularly shaped object (such as a wrench) is dropped from rest in a classroom and feels no air resistance, it will

A) accelerate but will not spin.
B) accelerate and turn until its center of gravity reaches its lowest point.
C) accelerate and turn about its center of gravity with uniform angular speed.
D) accelerate and turn about its center of gravity with uniform angular acceleration.
E) accelerate and spin until its center of gravity reaches its highest point.

Answer 2

accelerate but will not spin.

Question 3

A metal bar is hanging from a hook in the ceiling when it is suddenly struck by a ball that is moving horizontally (see figure). The ball is covered with glue, so it sticks to the bar. During this collision

A) the angular momentum of the system (ball and bar) is conserved about the hook because only gravity is acting on the system.
B) the angular momentum of the system (ball and bar) is not conserved because the hook exerts a force on the bar.
C) the angular momentum of the system (ball and bar) is conserved about the hook because neither the hook nor gravity exerts any torque on this system about the hook.
D) both the angular momentum of the system (ball and bar) and its kinetic energy are conserved.
E) both the linear momentum and the angular momentum of the system (ball and bar) are conserved.

Answer 3

the angular momentum of the system (ball and bar) is conserved about the hook because neither the hook nor gravity exerts any torque on this system about the hook.

Question 4

When a rigid body rotates about a fixed axis, all the points in the body have the same

A) tangential speed.
B) angular acceleration.
C) tangential acceleration.
D) linear displacement.
E) centripetal acceleration.

Question 5

As you are leaving a building, the door opens outward. If the hinges on the door are on your right, what is the direction of the angular velocity of the door as you open it?

A) up
B) down
C) to your left
D) to your right
E) forwards

Question 6

There must be equal amounts of mass on both side of the center of mass of an object.

A) True
B) False

Question 7

If and are nonzero vectors for which ∙ = 0, it must follow that

A) × = 0.
B) is parallel to .
C) | × | = AB.
D) | × | = 1.

Question 8

If two vectors point in opposite directions, their cross product must be zero.

A) True
B) False

Question 9

The angular momentum of a system remains constant

A) when the total kinetic energy is constant.
B) when no net external force acts on the system.
C) when the linear momentum and the energy are constant.
D) when no torque acts on the system.
E) all the time since it is a conserved quantity.

Question 10

A dumbbell-shaped object is composed by two equal masses, m, connected by a rod of negligible mass and length r. If I₁ is the moment of inertia of this object with respect to an axis passing through the center of the rod and perpendicular to it and I₂ is the moment of inertia with respect to an axis passing through one of the masses, it follows that

A) I₁ = I₂.
B) I₁ > I₂.
C) I₂ > I₁.

Question 11

A tire is rolling along a road, without slipping, with a velocity v. A piece of tape is attached to the tire. When the tape is opposite the road (at the top of the tire), its velocity with respect to the road is

A) 2v.
B) v.
C) 1.5v.
D) zero.
E) The velocity depends on the radius of the tire.

Question 12

A horizontal disk rotates about a vertical axis through its center. Point P is midway between the center and the rim of the disk, and point Q is on the rim. If the disk turns with constant angular velocity, which of the following statements about it are true? (There may be more than one correct choice.)

A) P and Q have the same linear acceleration.
B) Q is moving twice as fast as P.
C) The linear acceleration of Q is twice as great as the linear acceleration of P.
D) The linear acceleration of P is twice as great as the linear acceleration of Q.
E) The angular velocity of Q is twice as great as the angular velocity of P.

Question 13

When you ride a bicycle, in what direction is the angular velocity of the wheels?

A) to your left
B) to your right
C) forwards
D) backwards
E) up

Question 14

If the torque on an object adds up to zero

A) the forces on it also add up to zero.
B) the object is at rest.
C) the object cannot be turning.
D) the object could be accelerating linearly but it could not be turning.
E) the object could be both turning and accelerating linearly.

Question 15

The two rotating systems shown in the figure differ only in that the two identical movable masses are positioned at different distances from the axis of rotation. If you release the hanging blocks simultaneously from rest, and if the ropes do not slip, which block lands first?

A) The block at the left lands first.
B) The block at the right lands first.
C) Both blocks land at the same time.

Question 16

A solid sphere, solid cylinder, and a hollow pipe all have equal masses and radii and are of uniform density. If the three are released simultaneously at the top of an inclined plane and roll without slipping, which one will reach the bottom first?

A) solid sphere
B) hollow pipe
C) solid cylinder
D) They all reach the bottom at the same time.

Question 17

A ball is released from rest on a no-slip surface, as shown in the figure. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in the figure. When the ball reaches its maximum height on the frictionless surface, it is

A) at a greater height than when it was released.
B) at a lesser height than when it was released.
C) at the same height as when it was released.
D) It is impossible to tell without knowing the mass of the ball.
E) It is impossible to tell without knowing the radius of the ball.

Question 18

A uniform disk, a uniform hoop, and a uniform solid sphere are released at the same time at the top of an inclined ramp. They all roll without slipping. In what order do they reach the bottom of the ramp?

A) disk, hoop, sphere
B) hoop, sphere, disk
C) sphere, disk, hoop
D) sphere, hoop, disk
E) hoop, disk, sphere

Question 19

If two forces of equal magnitude act on an object that is hinged at a pivot, the force acting farther from the pivot must produce the greater torque about the pivot.

A) true
B) false
C) unable to decide without knowing the shape of the object

Question 20

If two vectors are perpendicular to each other, their cross product must be zero.

A) True
B) False

Question 21

When a 2.75-kg fan, having blades 18.5 cm long, is turned off, its angular speed decreases uniformly from 10.0 rad/s to 6.30 rad/s in 5.00 s.
(a) What is the magnitude of the angular acceleration of the fan?
(b) Through what angle (in degrees) does it turn while it is slowing down during the 5.00 s?
(c) If its angular acceleration does not change, how long after it is turned off does it take the fan to stop.

Question 22

A 1.15-kg grinding wheel 22.0 cm in diameter is spinning counterclockwise at a rate of 20.0 revolutions per second. When the power to the grinder is turned off, the grinding wheel slows with constant angular acceleration and takes 80.0 s to come to a rest.
(a) What was the angular acceleration (in rad/s²) of the grinding wheel as it came to rest if we take a counterclockwise rotation as positive?
(b) How many revolutions did the wheel make during the time it was coming to rest?

Question 23

A 2.00-m rod of negligible mass connects two very small objects at its ends. The mass of one object is 1.00 kg and the mass of the other is unknown. The center of mass of this system is on the rod a distance 1.60 m from the 1.00-kg mass object. What is the mass of the other object?

A) 4.11 kg
B) 3.22 kg
C) 4.00 kg
D) 0.250 kg
E) 0.800 kg

Question 24

A 3.45-kg centrifuge takes 100 s to spin up from rest to its final angular speed with constant angular acceleration. A point located 8.00 cm from the axis of rotation of the centrifuge moves with a speed of 150 m/s when the centrifuge is at full speed.
(a) What is the angular acceleration (in rad/s²) of the centrifuge as it spins up?
(b) How many revolutions does the centrifuge make as it goes from rest to its final angular speed?

Question 25

A uniform piece of wire, 20 cm long, is bent in a right angle in the center to give it an L-shape. How far from the bend is the center of mass of the bent wire?

A) 2.5 cm
B) 3.5 cm
C) 4.5 cm
D) 5.0 cm
E) 7.1 cm

Question 26

If = -4 - 2 - 3 , what is × ?

A) +3 - 4
B) +3 + 4
C) -3 + 4
D) +3 + 2 - 4
E) -3 - 2 + 4

Question 27

A 4.50-kg wheel that is 34.5 cm in diameter 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?

A) 0.616 rad/s²
B) 5.45 rad/s²
C) 111 rad/s²
D) 22.5 rad/s²
E) 10.9 rad/s²

Question 28

What is the magnitude of the cross product of a vector of magnitude 2.00 m pointing east and a vector of magnitude 4.00 m pointing 30.0° west of north?

A) 6.93
B) -6.93
C) 4.00
D) -4.00
E) 8.00

Question 29

What is the vector product of = 2.00 + 3.00 + 1.00 and = 1.00 - 3.00 - 2.00 ?

A) -3.00 + 5.00 - 9.00
B) -5.00 + 2.00 - 6.00
C) -9.00 - 3.00 - 3.00
D) -4.00 + 3.00 - 1.00
E) 2.00 -9.00 - 2.00

Question 30

A turbine blade rotates with angular velocity ω(t) = 2.00 rad/s- 2.1.00 rad/s³ . What is the angular acceleration of the blade at

A) -38.2 rad/
B) -19.1 rad/
C) -86.0 rad/
D) -36.2 rad/
E) -172 rad/

Question 31

If the magnitude of the cross product of two vectors is one-half the dot product of the same vectors, what is the angle between the two vectors?

Question 32

In the figure, a 60-cm length of uniform wire, of 60 g mass and negligible thickness, is bent into a right triangle. The x and y coordinates of the center of mass, in cm, are closest to

A) (8, 3).
B) (8, 5).
C) (9, 4).
D) (10, 3).
E) (10, 5).

Question 33

For the vectors shown in the figure, find the magnitude and direction of × , assuming that the quantities shown are accurate to two significant figures.

A) 26, directed into the plane
B) 26, directed out of the plane
C) 31, directed on the plane
D) 31, directed into the plane
E) 31, directed out of the plane

Question 34

In the figure, four point masses are placed as shown. The x and y coordinates of the center of mass are closest to

A) (2.2 m, 2.6 m).
B) (2.2 m, 2.7 m).
C) (2.3 m, 2.6 m).
D) (2.3 m, 2.7 m).
E) (2.3 m, 2.8 m).

Question 35

For the vectors shown in the figure, find the magnitude and direction of the vector product × , assuming that the quantities shown are accurate to two significant figures.

A) 16, directed into the plane
B) 16, directed out of the plane
C) 45, directed on the plane
D) 45, directed into the plane
E) 45, directed out of the plane

Question 36

A heavy boy and a lightweight girl are balanced on a massless seesaw. If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw? Assume that both people are small enough compared to the length of the seesaw to be thought of as point masses.

A) It is impossible to say without knowing the masses.
B) It is impossible to say without knowing the distances.
C) The side the boy is sitting on will tilt downward.
D) Nothing will happen; the seesaw will still be balanced.
E) The side the girl is sitting on will tilt downward.

Question 37

The angular acceleration of a wheel is given in by where t is in seconds. If the wheel starts from rest at when is the next time the wheel is at rest?

A) 5.1 s
B) 8.4 s
C) 6.9 s
D) 3.6 s

Question 38

The angular velocity of a 755-g wheel 15.0 cm in diameter is given by the equation ω(t) = (2.00 rad/s²)t + (1.00 rad/s⁴)t³.
(a) Through how many radians does the wheel turn during the first 2.00 s of its motion?
(b) What is the angular acceleration (in rad/s²) of the wheel at the end of the first 2.00 s of its motion?

Question 39

A long thin rod of length L has a linear density λ(x) = Ax where x is the distance from the left end of the rod.
(a) How far is the center of mass of the rod from the left end of the rod?
(b) What is the mass of the rod?

Question 40

If = -2 - 6 + 2 and = -2 -2 - 3 , which of the following numbers is closest to the magnitude of × ?

A) 25
B) 21
C) 17
D) 13
E) 9

Question 41

A piece of thin uniform wire of mass m and length 3b is bent into an equilateral triangle. Find the moment of inertia of the wire triangle about an axis perpendicular to the plane of the triangle and passing through one of its vertices.

A) mb²
B) mb²
C) mb²
D) mb²
E) mb²

Question 42

A light triangular plate OAB is in a horizontal plane. Three forces, and act on the plate, which is pivoted about a vertical axes through point O. In the figure, ₂ is perpendicular to OB. Consider the counterclockwise sense as positive. The sum of the torques about the vertical axis through point O, acting on the plate due to forces F₁, F₂, and F₃, is closest to

A) 4.1 N ∙ m.
B) 5.4 N ∙ m.
C) -4.1 N ∙ m.
D) -5.4 N ∙ m.
E) zero.

Question 43

An extremely light rod 1.00 m long has a 2.00-kg mass attached to one end and a 3.00-kg mass attached to the other. The system rotates at a constant angular speed about a fixed axis perpendicular to the rod that passes through the rod 30.0 cm from the end with the 3.00-kg mass attached. The kinetic energy of the system is measured to be 100.0 J.
(a) What is the moment of inertia of this system about the fixed axis?
(b) What is the angular speed (in revolutions per second) of this system?

Question 44

In the figure, point P is at rest when it is on the x-axis. The linear speed of point P when it reaches the y-axis is closest to

A) 0.18 m/s.
B) 0.24 m/s.
C) 0.35 m/s.
D) 0.49 m/s.
E) 0.71 m/s.

Question 45

In the figure, point P is at rest when it is on the x-axis. The time t, when P returns to the original position on the x-axis, is closest to

A) 13 s.
B) 18 s.
C) 25 s.
D) 35 s.
E) 50 s.

Question 46

What is the torque about the origin on a particle located at = 3 m + 4 m - 2 m if a force = 5 N - 2 N + 3 N acts on the particle?

A) (8 + - 26 ) N ∙ m
B) (8 - 19 - 26 ) N ∙ m
C) (8 + 2 + ) N ∙ m
D) (16 - 19 - 26 ) N ∙ m
E) (8 - 2 + ) N ∙ m

Question 47

A slender uniform rod 100.00 cm long is used as a meter stick. Two parallel axes that are perpendicular to the rod are considered. The first axis passes through the 50-cm mark and the second axis passes through the 30-cm mark. What is the ratio of the moment of inertia through the second axis to the moment of inertia through the first axis?

A) I₂/I₁ = 1.5
B) I₂/I₁ = 1.7
C) I₂/I₁ = 1.9
D) I₂/I₁ = 2.1
E) I₂/I₁ = 2.3

Question 48

A 72.0-kg person pushes on a small doorknob with a force of 5.00 N perpendicular to the surface of the door. The doorknob is located 0.800 m from axis of the frictionless hinges of the door. The door begins to rotate with an angular acceleration of 2.00 rad/s². What is the moment of inertia of the door about the hinges?

A) 4.28 kg ∙ m²
B) 7.52 kg ∙ m²
C) 1.88 kg ∙ m²
D) 0.684 kg ∙ m²
E) 2.74 kg ∙ m²

Question 49

In the figure, two blocks, of masses 2.00 kg and 3.00 kg, are connected by a light string that passes over a frictionless pulley of moment of inertia 0.00400 kg · m² and radius 5.00 cm. The coefficient of friction for the tabletop is 0.300. The blocks are released from rest. Using energy methods, find the speed of the upper block just as it has moved 0.600 m.

A) 1.22 m/s
B) 5.44 m/s
C) 3.19 m/s
D) 1.95 m/s
E) 1.40 m/s

Question 50

A uniform solid sphere of mass M and radius R rotates with an angular speed ω about an axis through its center. A uniform solid cylinder of mass M, radius R, and length 2R rotates through an axis running through the central axis of the cylinder. What must be the angular speed of the cylinder so it will have the same rotational kinetic energy as the sphere?

A) 2ω/5
B) ω
C) 4ω/5
D) 2ω/
E) ω/

Question 51

While spinning down from 500.0 rpm to rest, a solid uniform flywheel does of work. If the radius of the disk is what is its mass?

A) 5.2 kg
B) 4.4 kg
C) 6.0 kg
D) 6.8 kg

Question 52

A 1.10-kg wrench is acting on a nut trying to turn it. The length of the wrench lies directly to the east of the nut. A force 150.0 N acts on the wrench at a position 15.0 cm from the center of the nut in a direction 30.0° north of east. What is the magnitude of the torque about the center of the nut?

A) 22.5 N ∙ m
B) 11.3 N ∙ m
C) 19.5 N ∙ m
D) 2250 N ∙ m
E) 1949 N ∙ m

Question 53

In the figure, a weightlifter's barbell consists of two identical uniform spherical masses each with radius 0.17 m and mass of 50 kg. The weights are connected by a 0.96-m uniform steel rod with a mass of 12 kg. Find the moment of inertia of the barbell about the axis through the center (see figure).

Question 54

A 95 N force exerted at the end of a torque wrench gives rise to a torque of 15 N ∙ m. What is the angle (assumed to be less than 90°) between the wrench handle and the direction of the applied force?

A) 18°
B) 14°
C) 22°
D) 25°

Question 55

A force = 3.00 N - 2.00 N acts at a location = 1.00 m + 2.00 m on an object. What is the torque that this force applies about an axis through the origin perpendicular to the xy-plane?

A) -1.00 N ∙ m
B) 7.00 N ∙ m
C) -3.00 N ∙ m
D) 5.00 N ∙ m
E) -8.00 N ∙ m

Question 56

A machinist turns the power on to a grinding wheel, which is at rest at time t = 0.00 s. The wheel accelerates uniformly for 10 s and reaches the operating angular velocity of The wheel is run at that angular velocity for 37 s and then power is shut off. The wheel decelerates uniformly at until the wheel stops. In this situation, the time interval of angular deceleration (slowing down) is closest to:

A) 17 s
B) 15 s
C) 19 s
D) 21 s
E) 23 s

Question 57

A futuristic design for a car is to have a large solid disk-shaped flywheel within the car storing kinetic energy. The uniform flywheel has mass 370 kg with a radius of 0.500 m and can rotate up to Assuming all of this stored kinetic energy could be transferred to the linear velocity of the car, find the maximum attainable speed of the car.

Question 58

A uniform solid sphere has a moment of inertia I about an axis tangent to its surface. What is the moment of inertia of this sphere about an axis through its center?

A) 1/7 I
B) 2/7 I
C) 2/5 I
D) 3/5 I
E) 7/5 I

Question 59

A 1.25-kg ball begins rolling from rest with constant angular acceleration down a hill. If it takes 3.60 s for it to make the first complete revolution, how long will it take to make the next complete revolution?

Question 60

At any angular speed, a certain uniform solid sphere of diameter D has half as much rotational kinetic energy as a certain uniform thin-walled hollow sphere of the same diameter when both are spinning about an axis through their centers. If the mass of the solid sphere is M, the mass of the hollow sphere is

A) 3/5 M.
B) 5/3 M.
C) 5/6 M.
D) 6/5 M.
E) 2 M.

Question 61

A torque of 12 N ∙ m is applied to a solid, uniform disk of radius 0.50 m, causing the disk to accelerate at What is the mass of the disk?

A) 17 kg
B) 13 kg
C) 8.5 kg
D) 4.3 kg

Question 62

A uniform solid sphere of mass 1.5 kg and diameter 30.0 cm starts from rest and rolls without slipping down a 35° incline that is 7.0 m long.
(a) Calculate the linear speed of the center of the sphere when it reaches the bottom of the incline.
(b) Determine the angular speed of the sphere about its center at the bottom of the incline.
(c) Through what angle (in radians) does this sphere turn as it rolls down the incline?
(d) Does the linear speed in (a) depend on the radius or mass of the sphere? Does the angular speed in (b) depend on the radius or mass of the sphere?

Question 63

A uniform solid 5.25-kg cylinder is released from rest and rolls without slipping down an inclined plane inclined at 18° to the horizontal. How fast is it moving after it has rolled 2.2 m down the plane?

A) 4.3 m/s
B) 5.2 m/s
C) 3.0 m/s
D) 3.7 m/s
E) 2.6 m/s

Question 64

In the figure, a mass of 31.77 kg is attached to a light string that is wrapped around a cylindrical spool of radius 10.0 cm and moment of inertia 4.00 kg · m². The spool is suspended from the ceiling, and the mass is then released from rest a distance 5.70 m above the floor. How long does it take to reach the floor?

A) 3.98 s
B) 3.83 s
C) 1.14 s
D) 5.59 s
E) 7.89 s

Question 65

A very thin horizontal, 2.00-m long, 5.00-kg uniform beam that lies along the east-west direction is acted on by two forces. At the east end of the beam, a 200-N force pushes downward. At the west end of the beam, a 200-N force pushes upward. What is the angular acceleration of the beam?

A) 240 rad/s² north
B) 1.33 × 10² rad/s² north
C) zero
D) 240 rad/s² south
E) 1.33 × 10² rad/s² south

Question 66

In an experiment, a student brings up the rotational speed of a piece of laboratory apparatus to 30.0 rpm. She then allows the apparatus to slow down uniformly on its own, and counts 240 revolutions before the apparatus comes to a stop. The moment of inertia of the apparatus is known to be 0.0850 kg∙m². What is the magnitude of the retarding torque on the apparatus?

A) 0.0425 N ∙ m
B) 0.159 N ∙ m
C) 0.0787 N ∙ m
D) 0.000278 N ∙ m
E) 0.0000136 N ∙ m

Question 67

A solid, uniform sphere of mass 2.0 kg and radius 1.7 m rolls from rest without slipping down an inclined plane of height 7.0 m. What is the angular velocity of the sphere at the bottom of the inclined plane?

A) 5.8 rad/s
B) 9.9 rad/s
C) 11 rad/s
D) 7.0 rad/s

Question 68

For the apparatus shown in the figure, there is no slipping between the cord and the surface of the pulley. The blocks have mass of 3.0 kg and 5.7 kg, and the pulley has a radius of 0.12 m and a mass of 10.3 kg. At the instant the 5.7 kg mass has fallen 1.5 m starting from rest, find the speed of each block. (Assume the pulley is in the shape of a uniform solid disk and has no friction in its axle.)

Question 69

In the figure, a very light rope is wrapped around a wheel of radius R = 2.0 meters and does not slip. The wheel is mounted with frictionless bearings on an axle through its center. A block of mass 14 kg is suspended from the end of the rope. When the system is released from rest it is observed that the block descends 10 meters in 2.0 seconds. What is the moment of inertia of the wheel?

Question 70

A uniform solid sphere is rolling without slipping along a horizontal surface with a speed of 5.50 m/s when it starts up a ramp that makes an angle of 25.0° with the horizontal. What is the speed of the sphere after it has rolled 3.00 m up the ramp, measured along the surface of the ramp?

A) 4.01 m/s
B) 8.02 m/s
C) 1.91 m/s
D) 2.16 m/s
E) 3.53 m/s

Question 71

The rotating systems shown in the figure differ only in that the two identical movable masses are positioned a distance r from the axis of rotation (left), or a distance r/2 from the axis of rotation (right). You release the hanging blocks simultaneously from rest, and call t_L the time taken by the block on the left and t_R the time taken by the block on the right to reach the bottom, respectively. The bar, pulley, and rope have negligible mass, the rope does not slip, and there is no friction in the axle of the pulley. Under these conditions

A) t_L = t_R.
B) t_L = t_R.
C) t_L = t_R.
D) t_L = 2t_R.
E) t_l = 4t_R.

Question 72

A thin cylindrical shell is released from rest and rolls without slipping down an inclined ramp that makes an angle of 30° with the horizontal. How long does it take it to travel the first 3.1 m?

A) 1.4 s
B) 1.1 s
C) 2.1 s
D) 1.6 s
E) 1.8 s

Question 73

A uniform solid cylindrical log begins rolling without slipping down a ramp that rises 28.0° above the horizontal. After it has rolled 4.20 m along the ramp, what is the magnitude of the linear acceleration of its center of mass?

A) 9.80 m/s²
B) 4.60 m/s²
C) 3.29 m/s²
D) 3.07 m/s²
E) 2.30 m/s²

Question 74

A uniform solid cylinder of radius R and a thin uniform spherical shell of radius R both roll without slipping. If both objects have the same mass and the same kinetic energy, what is the ratio of the linear speed of the cylinder to the linear speed of the spherical shell?

A) / 2
B) / 2
C)
D) 4 /
E) 4 / 3

Question 75

A uniform hollow spherical ball of mass 1.75 kg and radius 40.0 cm rolls without slipping up a ramp that rises at 30.0° above the horizontal. The speed of the ball at the base of the ramp is 2.63 m/s. While the ball is moving up the ramp, find
(a) the acceleration (magnitude and direction) of its center of mass and
(b) the friction force (magnitude and direction) acting on it due to the surface of the ramp.

Question 76

A very thin uniform rod, 2.40 m long and of weight 135 N, has a frictionless hinge at its lower end. It starts out vertically from rest and falls, pivoting about the hinge. Just as it has rotated through an angle of 55.0°, what is the downward acceleration of the end farthest from the hinge?

A) 5.02 m/s²
B) 8.03 m/s²
C) 9.80 m/s²
D) 12.0 m/s²
E) 19.6 m/s²

Question 77

A lawn roller in the form of a uniform solid cylinder is being pulled horizontally by a horizontal force B applied to an axle through the center of the roller, as shown in the figure. The roller has radius 0.65 meters and mass 51 kg and rolls without slipping. What magnitude of the force B is required to give the center of mass of the roller an acceleration of ?

Question 78

A wheel has a radius of 0.40 m and is mounted on frictionless bearings. A block is suspended from a rope that is wound on the wheel and attached to it (see figure). The wheel is released from rest and the block descends 1.5 m in 2.00 s without any slipping of the rope. The tension in the rope during the descent of the block is 20 N. What is the moment of inertia of the wheel?

A) 3.5 kg · m²
B) 3.7 kg · m²
C) 3.9 kg · m²
D) 4.1 kg · m²
E) 4.3 kg · m²

Question 79

A solid uniform sphere of mass 1.85 kg and diameter 45.0 cm spins about an axle through its center. Starting with an angular velocity of 2.40 rev/s, it stops after turning through 18.2 rev with uniform acceleration. The net torque acting on this sphere as it is slowing down is closest to

A) 0.00593 N ∙ m.
B) 0.0372 N ∙ m.
C) 0.0466 N ∙ m.
D) 0.0620 N ∙ m.
E) 0.149 N ∙ m.

Question 80

A string is wrapped around a pulley with a radius of 2.0 cm and no appreciable friction in its axle. The pulley is initially not turning. A constant force of 50 N is applied to the string, which does not slip, causing the pulley to rotate and the string to unwind. If the string unwinds 1.2 m in 4.9 s, what is the moment of inertia of the pulley?

A) 0.17 kg ∙ m²
B) 17 kg ∙ m²
C) 14 kg ∙ m²
D) 0.20 kg ∙ m²
E) 0.017 kg ∙ m²