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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.
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 ωᵈ in the counterclockwise direction until it stops. Which of the following could the student have used in order to approximate the initial angular velocity of the rotating disk?
A slow-motion camera that filmed the disk to determine the amount of time it took a particular point on the disk to make its first revolution after the disk was set into motion
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.
All four forces are exerted on the stick that is initially at rest. What is the angular momentum of the stick after 2.0s?
150 ᵏᵍ•ᵐ²⁄ₛ
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.
Which of the four forces, when exerted in the absence of the other three forces, will change the angular momentum of the stick at the smallest rate?
F₄
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 ½MR² 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
A rod of length 0.5m is placed on a horizontal surface. One end of the rod is connected to a pivot that will allow the rod to rotate around the pivot in the absence of frictional forces. A lump of clay is launched toward the free end of the rod at a known speed vᶜ. When the lump of clay strikes the free end of the rod, it sticks to the rod. The equation for the rotational inertia of the rod about the pivot is I=⅓Mℓ². Which of the following quantities, when used together, could a student measure in order to determine the change in angular momentum of the rod from when it was initially at rest to the instant in time when the rod has rotated 90° in the counterclockwise direction? Select two answers.
The mass of the rod
The tangential speed of the end of the rod after it has rotated 90° in the counterclockwise direction
Two small objects of mass m₀ 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
For Angular Momentum (L),
L₁ = L₂
For Angular Speed (ω),
ω₁ < ω₂
Two small objects of mass m₀ 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
Calculate the slope of a graph of the change in angular momentum versus the product of the net torque and the time interval and determine if the slope is equal to one.
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.
A motor rotates an axle that is connected to one end of a rod, as shown in the figure. Students may adjust the speed at which the motor causes the axle and rod to rotate. During an experiment, what essential quantity or quantities should be measured or determined in order for the students to determine the rod's change in angular momentum after 8s? Justify your selection.
The average net torque applied to the rod, because the average net torque is related to the change in angular velocity of the rod.
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?
2.50s
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.
One end of a string is attached to a 0.1kg object. The string is wrapped around a pulley of known rotational inertia and radius 0.5m 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 0.1kg 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 your selection.
The student is correct, because the linear acceleration of the 0.1kg 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.
In an experiment, a solid, uniform sphere is at rest on a horizontal surface. A net force is applied tangentially to the edge of the sphere that is the greatest horizontal distance away from the central axis of the sphere. The sphere begins to rotate from rest until the force is no longer applied after 3s. Frictional forces between the sphere and the horizontal surface are considered to be negligible. Which two of the following quantities, when used together, could a student measure to determine the change in angular momentum of the sphere from 0s to 3s? Select two answers.
Radius of the sphere
Average net force exerted on the sphere from 0s to 3s
A system consists of a horizontal rod and two small objects, as shown. The horizontal rod has a total length 2ₗ₀ and mass M, with rotational inertia Iᵣ=ᵐˡ²⁄₃. Each object has a mass m₀ and a rotational inertia Iₒ=m₀r² where r is the distance from the object to the axle. The objects are fixed to the rod at the positions shown in the figure. The system is initially at rest, and a net torque is exerted on the system until it rotates with an angular speed, ω₀ about the vertical axis shown, which is through the center of the rod. What is the change in the angular momentum of the system from when the system was at rest to the instant when the net torque is no longer applied to the system?
(ᵐ₀ˡ²⁄₃+m₀(³⁴ˡ₀²⁄₂₅))ω₀
v v v v v v v
(Ml₀²/3 + m0(34l₀²/25)) * ω0
A student conducts an experiment in which the angular velocity of a rotating object about a central axis of rotation changes as a function of time, as shown by the graph. The student makes the following claim.
"The net torque responsible for the rotation of the object changes direction at approximately 5.0s."
Which of the following statements is correct about the student's evaluation of the data from the graph? Justify your selection.
The student is incorrect, because the angular acceleration of the object remains constant.
A student conducts an experiment in which one end of a rod may freely rotate around a pivot that is attached to its other end. The student collects the necessary data to construct a graph of the rod's angular momentum as a function of time, as shown. The student makes the following claim.
"This graph shows that the magnitude of the net torque exerted on the rod increases as time increases."
Which of the following statements is correct about the student's evaluation of the data from the graph? Justify your selection.
The student is incorrect, because the graph shows that the net torque exerted on the rod is constant as time increases.
In experiment 1, a disk of mass M₀ and radius R₀ rotates about a center axle, as shown in the figure. An object of mass m₀ hangs from a string that is wound around the disk. The object is released from rest and falls a vertical distance H before striking the ground. The string does not completely unwind. The disk's angular momentum is L₁ when the falling object strikes the ground. The rotational inertia of the disk is Iᵈ=ᵐʳ²⁄₂. In experiment 2, a thin circular hoop with mass M₀ and radius R₀ replaces the disk, and the experiment is repeated. The rotational inertia of the hoop is Iₕ=MR². How does the magnitude of the hoop's final angular momentum compare to the magnitude of the disk's final angular momentum?
The hoop's final angular momentum is greater than the disk's final angular momentum.
