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At time t = 0, a disk starts from rest and begins spinning about its center with a constant angular acceleration of magnitude α. At time tf, the disk has angular speed ωf. which of the following expressions correctly compares the final angular displacement θf of the disk at time tf to the angular displacement θ1/2 at time tf/2?
θf=4θ1/2
An object rotates with an angular speed that varies with time, as shown in the graph. How can the graph be used to determine the magnitude of the angular acceleration α of the object? Justify your selection.
Determine the area bounded by the line and the horizontal axis from 0 to 2s because the slope represents Δω/Δt
A graph of the angular velocity ω as a function of time t is shown for an object that rotates about an axis. Three time intervals, 1-3, are shown. Which of the following correctly compares the angular displacement Δθ of the object during each time interval?
Δθ2>Δθ1=Δθ3
The graph shows the angular velocity ω as a function of time t for a point on a rotating disk. The magnitude of the angular acceleration of the disk at t=2s is most nearly
1.5 rad/s^2
A student is provided with the data shown in the table for the angular speed of a wheel. Does the student have enough information to calculate the average angular acceleration of the wheel? Why or why not?
No, because the initial direction of rotation must be known to calculate the change in angular velocity.
The graph represents the angular velocity of a fan as a function of time. What is the total angular displacement of the fan during the time shown on the graph?
ω1(t2-t1/t2)
A thin rod that is free to rotate about its top end is initially held in place at an angular position -θi as shown. It is released, and after a time t it is at angular position θi. What is the average angular speed of the rod while it swings from -θi to θi?
2θi/t
A sphere is spinning with initial angular speed ω1 when it begins slowing at a constant angular acceleration of magnitude α1. After rotation through an angular displacement of Δθ1 ,the sphere comes to a rest. A second identical sphere has an initial angular speed ω2 when it begins slowing at a constant angular acceleration of magnitude 2α1 . The second sphere rotates through the same angular displacement Δθ1 before coming to a rest. Which of the following correctly compares ω1 to ω2?
ω2=ω1
The graph shows the angular velocity of a rotational wheel as a function of time. A student analyzing the graph claims that the wheel has a constant angular acceleration. Is the claim correct? Why or why not?
Yes, because the graph is linear.
Pulleys X and Y are each attached to a block by a string that wraps around the pulley. Both blocks are released and have the same linear acceleration α. As the blocks fall, the pulleys rotate about their centers. Pulley Y has a larger radius than pulley X. How does the angular acceleration αx of pulley X compare to the angular acceleration αy of pulley Y?
αx > αy
A ball of radius R starts from rest and rolls without slipping down a ramp in a time t, reaching a final speed vf at the bottom of the ramp. Which of the following is a correct expression for the angular acceleration of the ball as it rolls down the ramp?
Vf/tR
Wheels A and B are connected by a moving belt and are both free to rotate around their centers, as shown. The belt fíes bit slip on the wheels. The radius of wheel B is twice the radius of A. Wheel A has constant angular speed ωA and wheel B has constant angular speed ωB. Which of the following correctly relates ωA and ωB?
ωA = ωB
A solid, uniform cone is spinning with constant angular speed about a vertical axis through its center of mass, as shown. A small insect is initially standing in the surface of the cone at point 1. At a time, the insect has moved to point 2. Which of the following statements about the motion of the insect is true?
The linear speed of the insect is greater at point 1 than point 2.
At the instant shown in the figure, the disk is spinning abut an axle at its center and the angular speed of the disk is increasing at a constant rate. Points X and Y are labeled, with point Y being farther from the rotational axis than point X. Which of the following correctly compares the angular speed and the linear speed of points X and Y at this instant?
ωx = ωy; vx < vy
A horizontal disk is free to rotate about an axis through its center. Point P is located on the disk a distance x .
from the center. At time t = 0, the disk begins spinning from rest with constant angular acceleration. At time tr, the
disk has rotated through an angle 01. Which of the following is a correct expression for the instantaneous linear speed of Point P at time ti?
2x01/t1
Two people are sitting on a large metal disk at a park. Person 1 is halfway between the center and the edge, while Person 2 is at the edge of the disk, as shown. The disk accelerates from rest until it reaches an angular velocity of w. Which of the following correctly compares the angular acceleration, a, and the tangential acceleration, a, of Person 1 and Person 2?
B
A horizontal disk has an axle of radius r at its center, as shown. A string is wound around the axle and pulled at a constant speed v, causing the disk to rotate with angular speed c. The string does not slip on the axle.
If the axle is replaced with an axle with a radius larger than r, with what speed will the string need to be pulled so that the disk again rotates with angular speed w?
Greater than v
A disk of radius R is mounted on a fixed axis through its center. At time t = 0 it begins accelerating from rest
with a constant acceleration, reaching an angular speed a at a time t. Which one of the following is a correct expression for the radial component of the disk's linear acceleration at time t /2?
(R/4)w^2
One end of a string is attached to a 0.1 kg object. The string is wrapped around a pulley of known rotational inertia and radius 0.5 m that may rotate about its central axis. The central axis is supported by strings that are connected to the ceiling, as shown in Figure 1. In an experiment, the 0.1 kg object is released from rest, and the necessary data are collected to graph the distance fallen by the object as a function of the square of the time fallen, as shown in Figure 2.
A student makes the following claim.
"Figure 1 and Figure 2 can be used to determine the magnitude and direction of the net torque exerted on the pulley."
Which of the following statements is correct about the student's evaluation of the data from the graph? Justify our selection.
The student is correct, because the linear acceleration of the 0.1 kg object can be determined from the graph.
The angular acceleration of the pulley can then be determined. The direction of the net torque will be in the direction of the angular acceleration.
A uniform ladder of mass and length rests against a smooth wall at an angle 0 is the torque due to the weight of the ladder about its base?
MgLcos(0o)/2
The figure below represents a stick of uniform density that is attached to a pivot at the right end and has equally spaced marks along its length. Any one or a combination of the four forces shown can be exerted on the stick as indicated.
Two of the four forces are exerted on the stick. Which of the following predictions is correct about the change in angular velocity of the stick per unit of time?
When F1 and F2 are exerted on the stick, the stick will have the greatest change in angular velocity per unit of time.
A solid metal bar is at rest on a horizontal frictionless surface. It is free to rotate about a vertical axis at the left end. The figures below show forces of different magnitudes that are exerted on the bar at different locations. In which case does the bar's angular speed about the axis increase at the fastest rate?
A
The figure below represents the top view of a bar that can pivot about point Q. The arrows represent forces of equal magnitude that can be exerted near the ends of the bar in the directions shown. Which of the following forces, when exerted together, would produce a torque about point
Q with the greatest magnitude
4 and 5
A meteoroid is in a circular orbit 600 km above the surface of a distant planet. The planet has the same mass as Earth but has a radius that is 90% of Earth's (where Earth's radius is approximately 6370 km).
Does the planet exert a torque on the meteoroid with respect to the center of mass of the planet? Why or why not?
No, because the planet exerts a force on the meteoroid parallel to its position vector relative to the center of mass of the planet.
The figure above shows a uniform beam of length L and mass M that hangs horizontally and is attached to a vertical wall. A block of mass M is suspended from the far end of the beam by a cable. A support cable runs from the wall to the outer edge of the beam. Both cables are of negligible mass. The wall exerts a force Fw on the left end of the beam.
For which of the following actions is the magnitude of the vertical component of Fw smallest?
D. Moving both the support cable and the block to the center of the beam.
The figure above shows a uniform meterstick that is set on a fulcrum at its center. A force of magnitude F toward the bottom of the page is exerted on the meterstick at the position shown. At which of the labeled positions must an upward force of magnitude 2F be exerted on the meterstick to keep the meterstick in equilibrium?
A. A
B. B
C. C
D. D
B. B
A uniform meterstick is balanced at the center, as shown below.
Which of the following shows how a 0.50 kg mass and a 1.0 kg mass could be hung on the meterstick so that the stick stays balanced?
B.
A uniform horizontal beam of mass M and length L is attached to a hinge at point P, with the opposite end supported by a cable, as shown in the figure. The angle between the beam and the cable is θ₀.
What is the magnitude of the torque that the cable exerts on the beam?
A. MgL₀/2
B. MgL₀
C. MgL₀sin(θ₀)/2
D. MgL₀sin(θ₀)
A. MgL₀/2
A wheel of radius R and negligible mass is mounted on a horizontal frictionless axle so that the wheel is in a vertical plane. Three small objects having masses m, M, and 2M, respectively, are mounted on the rim of the wheel, as shown below.
If the system is in static equilibrium, what is the value of m in terms of M?
A. M/2
B. M
C. 3M/2
D. 5M/2
C. 3M/2
A massless rigid rod of length 3d is pivoted at a fixed point W, and two forces each of magnitude F are applied vertically upward as shown below.
A third vertical force of magnitude F may be applied, either upward or downward, at one of the labeled points. With the proper choice of direction at each point, the rod can be in equilibrium if the third force of magnitude F is applied at point ____.
A. W only
B. Y only
C. V or X only
D. V or Y only
C. V or X only
A horizontal, uniform board of weight 125 N and length 4 m is supported by vertical chains at each end. A person weighing 500 N is sitting on the board. The tension in the right chain is 250 N. How far from the left end of the board is the person sitting?
A. 0.4 m
B. 1.5 m
C. 2 m
D. 2.5 m
B. 1.5 m
An object weighing 120 N is set on a rigid beam of negligible mass at a distance of 3 m from a pivot, as shown above. A vertical force is to be applied to the other end of the beam a distance of 4 m from the pivot to keep the beam at rest and horizontal.
What is the magnitude F of the force required?
A. 10 N
B. 30 N
C. 90 N
D. 120 N
C. 90 N
A rectangular block of mass m, width w, and height h is attached to a tabletop by a hinge at the block's lower right corner, as shown. To hold the block in place, a force of magnitude F is exerted perpendicular to the block at a distance y above the bottom of the left edge of the block. The block is in rotational equilibrium.
Which of the following expressions is equal to the distance y?
A. wmg/2F
B. wmg/F
C. hmg/2F
D. hmg/F
A. wmg/2F
Static friction in the hinges of a classroom door can exert a maximum total torque on the door of 100 N m. A student can push on the door with a force of 200 N. What is the maximum perpendicular distance from the hinges that the student can push on the door without the door moving?
A. 0.005 m
B. 0.01 m
C. 0.5 m
D. 2.0 m
C. 0.5 m
A disk is initially at rest on a tabletop when 4 forces, all with the same magnitude, are briefly exerted on the disk at the locations and in the directions shown in the bottom-view diagram. There is negligible friction between the disk and the tabletop.
Which of the following correctly describes the motion of the disk immediately after the forces have been exerted?
A. The disk will rotate at a constant angular speed.
Before graphs show the angular velocity ω of identical disks as a function of time t over the same time interval.
In which of the graphs is the disk in rotational equilibrium?
A. Graph 1 only
B. Graphs 1 and 2
C. Graph 3 only
D. Graphs 3 and 4
B. Graphs 1 and 2
The uniform rod shown in the figure is fixed to a table by a pivot through its center. The rod is free to rotate about its center. The left end of the rod is at position x=0 and the right is at x=L. A force of magnitude F is exerted at position X=3/4 L and directed toward the top of the page.
At what position would an additional force of magnitude 2F directed toward the top of the page need to be exerted so that the rod is in rotational equilibrium?
A. x=1/8L
B. x=1/4L
C. x=3/8L
D. x=1/2L
C. x = 3/8L
A bar is attached to a pivot at one end of the bar, as shown, and initially held in place in a horizontal orientation above the ground. The bar is then released and swings with negligible friction about the pivot. The bar passes through the labeled orientations A, B, and C before momentarily coming to rest at orientation D.
In which of the following orientations, if any, is the bar and rotational equilibrium about the pivot?
A. A & C
B. B only
C. D only
D. None of the orientations.
B. B only
A meterstick of negligible mass is free to rotate about an axis at its left end and is in rotational equilibrium. In addition to a force exerted at the axis of rotation, three forces are exerted on the meterstick. Two of the forces are indicated in the figure. The third force is exerted at the dot at the midpoint of the meterstick.
The magnitude of the torque exerted by the third force is most nearly ___.
A. 5 N⋅m
B. 15 N⋅m
C. 25 N⋅m
D. 40 N⋅m
B. 15 N⋅m
A disk of known radius and rotational inertia can rotate without friction in a horizontal plane around its fixed central axis. The disk has a cord of negligible mass wrapped around its edge. The disk is initially at rest, and the cord can be pulled to make the disk rotate. Which of the following procedures would best determine the relationship between applied torque and the resulting change in angular momentum of the disk?
D. For five forces of different magnitudes, pulling on the cord for 5 s, and then measuring the final angular velocity of the disk.
The axle of a wheel with rotational inertia 0.25 kg⋅m² around the center of the wheel is attached to a motor. A brake can be applied at the rim of the wheel at a distance of 0.5 m from the axle, as shown. With the motor exerting a constant torque on the wheel, the brake exerts a force of 2.0 N on the wheel, causing the wheel to rotate with a constant angular speed of 3.0 rad/s.
The magnitude of the torque exerted by the motor is most nearly ___.
A. 0
B. 0.25 N⋅m
C. 1.0 N⋅m
D. 2.0 N⋅m
C. 1.0 N⋅m
The figure shows a rod that is fixed to a horizontal surface at pivot P. The rod is initially rotating without friction in the counterclockwise direction. At time t, three forces of equal magnitude are applied to the rod as shown.
Which of the following is true about the angular speed and direction of rotation of the rod immediately after time t?
A. Angular Speed - Decreasing
Direction of Rotation - Counterclockwise
A disk with radius of 0.5 m is free to rotate about its center without friction. A string wrapped around the disk is pulled, as shown below, exerting a 2 N force tangent to the edge of the disk for 1 s.
If the disk starts from rest, what is its angular speed after 1 s?
A. 0
B. 1 rad/s
C. 4 rad/s
D. It cannot be determined without knowing the rotational inertia of the disk.
D. It cannot be determined without knowing the rotational inertia of the disk.
An axle passes through a pulley. Each end of the axle has a string that is tied to a support. A third string is looped many times around the edge of the pulley and the free end attached to a block of mass mB, which is held at rest. When the block is released, the block falls downward. Consider clockwise to be the positive direction of rotation, frictional effects from the axle are negligible, and the string wrapped around the disk never fully unwinds. The rotational inertia of the pulley is 1/2MR² about its center of mass.
Which of the following graphs, if any, shows the angular velocity of the pulley as a function of time after the block is released from rest?
B.
One end of a string of negligible mass is wrapped around a pulley and the other end is connected to a hanging block of mass m. A motor exerts a torque of magnitude τ on the pulley and causes it to rotate in the clockwise direction, lifting the block. The radius of the pulley is R and the rotational inertia of the pulley is I.
Which of the following equations accurately represents this scenario and could be used to solve for the angular acceleration α of the pulley?
B. τ - m(αR + g)R = Iα
A uniform metal pole is fixed to an axle at center and has two blocks attached to it. Each block is free to slide along the pole and the blocks are initially placed as shown in the diagram. A string is wrapped around the axle and connected over a pulley to a motor, as shown. The motor pulls the string downward, causing the axle and the pole to rotate. As the pole rotates, the blocks begin to slide apart toward the end of the pole. There is negligible friction between the blocks and the pole. As the blocks are sliding, the angular acceleration of the axle and pole is constant.
Which of the following correctly describes how the magnitude of the force exerted by the motor on the string changes, if at all, as the blocks slide apart?
D. The force is constant the entire time
A uniform bar is initially at rest on a horizontal surface when the two forces indicated in the top-view figure are exerted at the ends of the rod. The bar is free to rotate about a fixed pivot at the bar's center that anchors the center of the bar to the surface. Friction forces are negligible. As indicated by the arrows in the figure, F₁>F₂.
A student claims that at the instant shown in the figure the pivot is exerting a nonzero force on the bad and exerting a nonzero torque on the bar about the pivot. Is the student's claim correct? Why or why not?
D. No, because frictional forces are negligible, and therefore the bar slides freely on the pivot and the pivot exerts no force or torque on the bar.
In Figure 1, a hoop is spinning in place with constant angular velocity ω on a smooth surface with negligible friction. In Figure 2, the same hoop is rolling without slipping to the right with constant angular velocity ω on a rough surface with friction is not negligible.
Which of the following correctly compares the net torque and net force exerted on the hoop in Figures 1 and 2?
D. The net force and net torque are both zero in Figure 1, but the net torque and net force are both nonzero in Figure 2.
As shown in the top view diagram, a person exerts a perpendicular force of magnitude F on a door at a distance d from the hinge. The length of the door is L, and the door has rotational inertia I about the hinge. The door rotates about the hinge with negligible friction.
Which of the following expressions is equal to the tangential acceleration of the right end of the door?
A. dF/I
B. d²F/I
C. LFd/I
D. L²F/I
C. LFd/I
The disk shown in the top-view figures is on a horizontal surface and is free to move with negligible friction. In both figures, two forces of magnitude F, are exerted on the disk at the locations and in the directions shown.
Which of the following correctly indicates whether the net torque exerted on the disk about its center of mass is greater in Figure 1 or Figure 2, and provides a valid justification?
A. Figure 1, because both forces exert a clockwise torque.
A disk with mass M, radius R, and rotational inertia I = 1/2MR² around its center is free to rotate with negligible friction forces about a horizontal axle at its center, as shown. A string with negligible mass is wrapped around the disk multiple times and a small block with mass m is attached to its free end.
The block is released and the disk begins to rotate. Which of the following expressions is equal to the tension in the string after the block is released?
B. (m + M/2)g
A uniform disk, Disk A, of mass m and radius R rotates about an axis through the disk's geometric center as shown in Figure 1. The rotational inertia for the disk in this configuration is I = 1/2mR². An identical disk, Disk B, rotates about an axis that passes through the edge of the disk, a radial distance R from the center of the disk as shown in Figure 2. The same torque is exerted on both disks, and disks A and B experience angular acceleration αA and αB respectively.
Which of the following is a correct expression for the ratio of the two angular accelerations, αB/αA?
A. 1/3
