Deck 10: Dynamics of Rotational Motion

Question 1

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.

Answer 1

t_L = 2t_R.

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 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

Answer 3

sphere, disk, hoop

Question 4

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 than 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 5

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 6

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 7

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 8

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 9

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.

Question 10

What is the torque about the origin on a particle located at = 3 m + 4 m - 2 m if a force 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 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 95 N force exerted at the end of a torque wrench gives rise to a torque of 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 13

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 ,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

Question 14

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 15

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 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 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 18

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.

Question 19

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 20

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 21

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 22

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 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 23

Three solid,uniform,cylindrical flywheels,each of mass 65.0 kg and radius 1.47 m rotate independently around a common axis.Two of the flywheels rotate in one direction at the other rotates in the opposite direction at 3.42 rad/s.Calculate the magnitude of the net angular momentum of the system.

A) 298 kg • m²/s
B) 778 kg • m²/s
C) 257 kg • m²/s
D) 222 kg • m²/s

Question 24

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 25

A 500-g particle is located at the point = 4m + 3m - 2m and is moving with a velocity .What is the angular momentum of this particle about the origin?

A) (24 - 6 - 8 ) kg • m²/s
B) (12 - 3 - 4 ) kg • m²/s
C) (8 + 14 - 13 ) kg • m²/s
D) (10 - + 2 ) kg • m²/s
E) (4 - 13 - 11.5 ) kg • m²/s

Question 26

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 27

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 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 28

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 29

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 30

A bicycle is traveling north at 5.0 m/s.The mass of the wheel,2.0 kg,is uniformly distributed along the rim,which has a radius of 20 cm.What are the magnitude and direction of the angular momentum of the wheel about its axle?

A) 2.0 kg∙m²/s towards the west
B) 5.0 kg∙m²/s vertically upwards
C) 2.0 kg∙m²/s towards the east
D) 5.0 kg∙m²/s towards the east
E) 5.0 kg∙m²/s towards the west

Question 31

A potter's wheel,with rotational inertia 46 kg • m²,is spinning freely at 40 rpm.The potter drops a lump of clay onto the wheel,where it sticks a distance from the rotational axis.If the subsequent angular speed of the wheel and clay is 32 rpm what is the mass of the clay?

A) 8.0 kg
B) 5.4 kg
C) 7.0 kg
D) 8.8 kg

Question 32

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 33

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 34

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 35

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 36

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 37

A uniform solid disk of radius 1.60 m and mass 2.30 kg rolls without slipping to the bottom of an inclined plane.If the angular velocity of the disk is at the bottom,what is the height of the inclined plane?

A) 5.61 m
B) 4.21 m
C) 4.94 m
D) 6.73 m

Question 38

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 39

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 40

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 41

An object is rotating with an angular momentum 4.00 kg • m²/s while being acted on by a constant torque .What is the angular speed of precession of the angular velocity of the object?

A) 1.33 rad/s
B) 0.750 rad/s
C) 12.0 rad/s
D) zero
E) It depends on the moment of inertia of the object.

Question 42

A turntable has a radius of 0.80 m and a moment of inertia of 2.0 kg • m².The turntable is rotating with an angular velocity of 1.5 rad/s about a vertical axis though its center on frictionless bearings.A very small 0.40-kg ball is projected horizontally toward the turntable axis with a velocity of .The ball is caught by a very small and very light cup-shaped mechanism on the rim of the turntable (see figure).What is the angular velocity of the turntable just after the ball is caught?

A) 2.1 rad/s
B) 1.3 rad/s
C) 0.94 rad/s
D) 0.75 rad/s
E) 0.30 rad/s

Question 43

In the figure,the rotor of a gyroscope is a uniform disk that has a radius of 13.8 cm and a moment of inertia about its axis of .The length of the rotor shaft is 6.9 cm.The ball pivot at P is frictionless.At a given instant,the rotor shaft is horizontal and the rotor is rotating with an angular velocity of 90 rad/s about its axis OP,as shown.The rotor is viewed from point Q on the axis.The precessional linear velocity of point O,including the direction seen from point Q,is closest to

A) 0.054 m/s, leftward.
B) 0.054 m/s, upward.
C) 0.054 m/s, rightward.
D) 0.11 m/s, downward.
E) 0.11 m/s, rightward.

Question 44

A record is dropped vertically onto a freely rotating (undriven)turntable.Frictional forces act to bring the record and turntable to a common angular speed.If the rotational inertia of the record is 0.54 times that of the turntable,what percentage of the initial kinetic energy is lost?

A) 35%
B) 18%
C) 46%
D) 60%

Question 45

A turntable has a radius of 0.80 m and a moment of inertia of 2.0 kg • m².The turntable is rotating with an angular velocity of 1.5 rad/s about a vertical axis though its center on frictionless bearings.A very small 0.40-kg ball is projected horizontally toward the turntable axis with a velocity of 3.0 m/s.The ball is caught by a very small and very light cup-shaped mechanism on the rim of the turntable (see figure).The percent of the initial kinetic energy of the system that is lost during the capture of the ball is closest to

A) 45%.
B) 51%.
C) 55%.
D) 60%.
E) 65%.

Question 46

61)A bicycle wheel of radius 0.36 m and mass 3.2 kg is set spinning at 4.00 rev/s.A very light bolt is attached to extend the axle in length,and a string is attached to the axle at a distance of 0.10 m from the wheel.Initially the axle of the spinning wheel is horizontal,and the wheel is suspended only from the string.We can ignore the mass of the axle and spokes.At what rate will the wheel process about the vertical?

A) 2.9 rpm
B) 1.9 rpm
C) 18 rpm
D) 0.30 rpm
E) 0.77 rpm

Question 47

A figure skater rotating at 5.00 rad/s with arms extended has a moment of inertia of .If the arms are pulled in so the moment of inertia decreases to ,what is the final angular speed?

A) 2.25 rad/s
B) 4.60 rad/s
C) 6.25 rad/s
D) 1.76 rad/s
E) 0.810 rad/s

Question 48

A uniform disk has a mass of 3.7 kg and a radius of 0.40 m.The disk is mounted on frictionless bearings and is used as a turntable.The turntable is initially rotating at 30 rpm.A thin-walled hollow cylinder has the same mass and radius as the disk.It is released from rest,just above the turntable,and on the same vertical axis.The hollow cylinder slips on the turntable for 0.20 s until it acquires the same final angular velocity as the turntable.What is the final angular momentum of the system?

A) 0.93 /s
B) 1.1 /s
C) 1.3 /s
D) 1.6 /s
E) 1.9 /s

Question 49

A 5.0-m radius playground merry-go-round with a moment of inertia of 2000 kg • m² is rotating freely with an angular speed of 1.0 rad/s.Two people,each having a mass of 60 kg,are standing right outside the edge of the merry-go-round and step on it with negligible speed.What is the angular speed of the merry-go-round right after the two people have stepped on?

A) 0.20 rad/s
B) 0.40 rad/s
C) 0.60 rad/s
D) 0.80 rad/s
E) 0.67 rad/s