Rotational Motion

A wheel with radius 0.33 m and rotational inertia 2.0 kg⋅m2 spins on an axle with an initial angular speed of 3.0 rad/s. Friction in the axle exerts a torque on the wheel, causing the wheel to stop after 6.0 s. The average torque exerted on the wheel as it slows down has magnitude

1.0 N⋅m The average angular acceleration must first be determined by applying the correct angular kinematic equation. ω=ω0+αt

A horizontal disk of radius 0.2m and mass 0.3kg is mounted on a central vertical axle so that a student can study the relationship between net torque and change in angular momentum of the disk. In the experiment, the student uses a force probe to collect data pertaining to the net torque exerted on the edge of the disk as a function of time, as shown in the graph. The disk is initially at rest. At what instant in time does the disk have the greatest angular momentum?

2.5 s The area bound by the curve and the horizontal axis for a particular time interval for the graph of the net torque exerted on the disk as it rotates as a function of time represent the angular impulse exerted on the disk. Although the net torque exerted on the disk is zero at this time, a net torque has been exerted on the disk from 0 s0 s until 2.125s2.125s. The angular momentum of the disk increased during this entire interval of time. From 2.125s2.125s until 2.5s2.5s, the angular momentum of the disk remains unchanged.

The wheel on a vehicle has a rotational inertia of 2.0 kg⋅m2. At the instant the wheel has a counterclockwise angular velocity of 6.0 rad/s, an average counterclockwise torque of 5.0 N⋅m is applied, and continues for 4.0 s. What is the change in angular momentum of the wheel?

20 kg⋅m2/s

Planet X is in a stable circular orbit around a star, as shown in the figure. Which of the following graphs best predicts the angular momentum of the planet as a function of its horizontal position from point A to point B if the planet is moving counterclockwise as viewed in the figure above?

Flat positive line

A uniform ladder of mass M and length L rests against a smooth wall at an angle θ0, as shown in the figure. What is the torque due to the weight of the ladder about its base?

MgLcos(θ0)/2 T=rf perpendicular force

A student hangs a block from a light string that is attached to a massive pulley of unknown radius R, as shown in the figure. The student allows the block to fall from rest to the floor. Which two of the following sets of data that could be measured or determined should the student use together to determine the final angular velocity of the pulley just before the block hits the floor? Select two answers. Justify your selections.

The mass of the block, the distance of the block above the floor, and the amount of time it takes the block to reach the floor, because these quantities can be used to determine the acceleration of the block. The radius and mass of the pulley, because these quantities can be used together to determine the rotational inertia of the pulley.

An object revolves around a central axis of rotation. The motion of the object is described by the following equation. ω2=(10rad/s)2−(4rad/s2)θω Which two of the following graphs correctly shows the angular motion of the object? Select two answers.

exponential up or just straight down

A uniform horizontal beam of mass M and length L0 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?

(MgL)/2 Since the beam is uniform, the beam's weight can be modeled as acting downward perpendicular to and at the center of the beam. The force that the hinge exerts on the beam is unknown. Calculating the sum the individual torques about point P where the hinge is located allows for the torque exerted on the beam from the cable to be determined.

A disk-shaped platform has a known rotational inertia ID. The platform is mounted on a fixed axle and rotates in a horizontal plane with an initial angular velocity of ωD in the counterclockwise direction, as shown. After an unknown time interval, the disk comes to rest. A single point on the disk revolves around the center axle hundreds of times before the disk comes to rest. Frictional forces are considered to be constant. In a different experiment, the original disk is replaced with a disk for which frictional forces are considered to be negligible. The disk is set into motion such that it rotates with a constant angular speed. As the disk spins, a small sphere of clay is dropped onto the disk, and the sphere sticks to the disk. Which of the following claims is correct about the angular momentum and the total kinetic energy of the disk-sphere system immediately before and immediately after the collision?

Angular momentum stays the same, kinetic energy was greater before the collision Immediately before the collision and immediately after the collision, the disk-sphere system may be treated as a closed system in which no external forces are exerted on the system. Since no external forces are exerted on the system immediately before the collision, during the collision, and immediately after the collision, the angular momentum of the system does not change. However, the rotational collision under consideration is an inelastic collision. This conclusion can be made because the sphere sticks to the disk and travels with the disk at a common angular speed. In an inelastic collision, the kinetic energy of the system immediately before the collision is greater than the kinetic energy of the system immediately after the collision. Mechanical energy of the system is converted into nonmechanical energy during the collision.

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?

B

A rod is at rest on a flat, horizontal surface. One end of the rod is attached to a pivot, and the rod may freely rotate around the pivot if acted upon by a net external torque, as shown in Figure 1. In an experiment, the rod is initially at rest and student exerts a net torque on the rod. Data are collected to create a graph of the rod's angular acceleration as a function of time, as shown in Figure 2. Frictional forces are considered to be negligible. How can the student use the graph to determine the angular momentum of the rod at 5 s?

Determine the area bound by the curve and the horizontal axis from 0 s to 5 s and multiply the result by the rotational inertia of the rod. Analysis of this graph can be used in conjunction with the angular impulse-momentum theorem. The area bound by the curve and the horizontal axis for a particular time interval of an object's or point's angular acceleration as a function of time is equal to the change in the angular velocity of the object. If the change in the angular velocity is determined, then the change in the angular momentum of the rod is also determined of the change in angular velocity is multiplied by the rotational inertia of the rod.

Four forces are exerted on a disk of radius RR that is free to spin about its center, as shown above. The magnitudes are proportional to the length of the force vectors, where F1=F4 , F2=F3 , and F1=2F2. Which two forces combine to exert zero net torque on the disk? Select two answers.

F2 and F4 The torque exerted on an object by a force F is τ=r⊥F=rFsinθ. Since F1, F2, and F3 all exert torques in the same (counterclockwise) direction, no two of them can be combined to produce a net torque of zero. One of them must combine with the torque exerted by F4, the only force that exerts a clockwise torque, in order to produce zero net torque.

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?

For five forces of different magnitude, pulling on the cord for 5 s, and then measuring the final angular velocity of the disk

A 6.0 kg block initially at rest is pushed against a wall by a 100 N force as shown. The coefficient of kinetic friction is 0.30 while the coefficient of static friction is 0.50. What is true of the friction acting on the block after a time of 1 second?

Kinetic friction acts downward on the block

A lump of clay of mass mclay with speed vclay=8 m/s travels toward various spheres that are suspended from the ceiling by lightweight strings of different lengths, as shown in the figure. For the three scenarios, the clay collides with the suspended sphere and sticks to it. Which of the following correctly relates the angular momentum L of the clay-bob system immediately after the collision for each scenario, where the angular momentum is taken about the point where the string is attached to the ceiling?

L1>L2>L3 The conservation of angular momentum should be applied to this scenario. Therefore, the angular momentum of the clay immediately before the collision or the angular momentum of the clay-bob system immediately after the collision may be calculated to make the determination.The angular momentum of the clay immediately before the collision for scenario 11 can be determined.

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?

Moving both the support cable and the block to the center of the beam

A disk-shaped platform has a known rotational inertia ID. The platform is mounted on a fixed axle and rotates in a horizontal plane with an initial angular velocity of ωD in the counterclockwise direction, as shown. After an unknown time interval, the disk comes to rest. A single point on the disk revolves around the center axle hundreds of times before the disk comes to rest. Frictional forces are considered to be constant. A student must determine the angular impulse that frictional forces exert on the disk from the moment it rotates with angular velocity in the counterclockwise direction until it stops. What additional data, if any, should a student collect to determine the angular impulse on the disk? Justify your selection.

No additional data are necessary, because the rotational inertia of the disk and its initial angular velocity are known. An angular impulse can be mathematically defined by using the appropriate equation. ΔL=τΔt, where L=Iω The angular impulse under consideration is −IDωD.

The diagram above shows a top view of a child of mass M on a circular platform of mass 5M that is rotating counterclockwise. Assume the platform rotates without friction. Which of the following describes an action by the child that will result in an increase in the total angular momentum of the child-platform system?

None of the actions described will change the total angular momentum of the child-platform system. There are no external torques acting on the system because the child is included

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 12MR2 about its center of mass. The block falls for a time t0, but the string does not completely unwind. What is the change in angular momentum of the pulley-block system from the instant that the block is released from rest until time t0?

Rmbgt0 The change in angular momentum for the pulley-block system is the product of the net external torque mbgR and the time interval t0.

A disk is initially rotating counterclockwise around a fixed axis with angular speed w0. At time t = 0, the two forces shown in the figure above are exerted on the disk. If counterclockwise is positive, which of the following could show the angular velocity of the disk as a function of time?

Slope downward The net torque on the disk is constant and oriented clockwise. So, the disk has a constant clockwise angular acceleration. Because counterclockwise is positive, this angular acceleration — the rate of change of the angular velocity — is constant and negative.

Two small objects of mass m0m0 and a rotating platform of radius R and rotational inertia Ip about its center compose a single system. Students use the system to conduct two experiments. The objects are assumed to be point masses. Each object of mass m0 is placed a distance r1 away from the center of the platform such that both masses are on opposite sides of the platform. A constant tangential force F0 is applied to the edge of the platform for a time Δt0, as shown in Figure 1. The system is initially at rest Each object of mass m0 is placed a distance r2 away from the center of the platform such that both masses are on opposite sides of the platform. Distance r2<r1 . A constant tangential force F0 is applied to the edge of the platform for a time Δt0 , as shown in Figure 2. The system is initially at rest. Which of the following graphs represents the angular displacement of the system as a function of time for the system in experiment 1?

Slope exponential upward the slope of an angular position versus time graph represents the angular velocity of a point on an object that rotates. As time increases, the slope of this graph increases. This means that the angular speed of the system increases. Therefore, this graph represents the situation described for experiment 1.

A rod of length 2D0 and mass 2M0 is at rest on a flat, horizontal surface. One end of the rod is connected to a pivot that the rod will rotate around if acted upon by a net torque. A sphere of mass m0 is launched horizontally toward the free end of the rod with velocity v0, as shown in the figure. After the sphere collides with the rod, the sphere sticks to the rod and both objects rotate around the pivot with a common angular velocity. Which of the following predictions is correct about angular momentum and rotational kinetic energy of the sphere-rod system immediately before the collision and immediately after the collision?

The angular momentum immediately before the collision is equal to the angular momentum immediately after the collision. The rotational kinetic energy immediately before the collision is greater than the rotational kinetic energy immediately after the collision. The rotational collision under consideration is an inelastic collision in which no net torque is exerted on the system. Therefore, the angular momentum is the same immediately before the collision and immediately after the collision. However, in an inelastic collision, kinetic energy is converted into nonmechanical energy during the collision. Therefore, the rotational kinetic energy of the system immediately before the collision is greater than the rotational kinetic energy of the system immediately after the collision.

A disk of radius 50cm rotates about a center axle. The angular position as a function of time for a point on the edge of the disk is shown. Which two of the following quantities of the point on the edge of the disk can be correctly mathematically determined from the graph using the methods described? Justify your selections. Select two answers.

The angular velocity, because this quantity can be determined by calculating the slope of the graph. B The translational speed, because v=rω.

The diagram above shows a top view of a child of mass M on a circular platform of mass 2M that is rotating counterclockwise. Assume the platform rotates without friction. Which of the following describes an action by the child that will increase the angular speed of the platform-child system and gives the correct reason why?

The child moves toward the center of the platform, decreasing the rotational inertia of the system.

The figure above 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. T=rfsintheta

A graph of the angular velocity ωω 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 area bound by the curve and the horizontal axis for a particular time interval for a graph of an object's or point's angular velocity as a function of time is equal to the angular displacement of the object. Time interval 1 and time interval 3 are in the shape of a right triangle with the same base and height, so the angular displacement for both time intervals is the same. Time interval 2 is in the shape of a rectangle that has the same height as the right triangles that represent time interval 1 and time interval 2. Therefore, the area of the rectangle is greater than the area of the two right triangles. The greatest angular displacement takes place during time interval 2.