AP Physics 1 Unit 7 Progress Check A + B
A merry-go-round disk with a rotational inertia of Id about its center spins around its center axle with an initial angular velocity of ω0. A child is standing near the edge of the merry-go-round, as shown in Figure 1. The child's rotational inertia about the center of the disk when near the edge is I0. As the merry-go-round spins, the child moves closer to the center, as shown in Figure 2, until the disk rotates with an angular velocity of ω1. Which of the following equations could a student use to determine the rotational inertia Is of the child-merry-go-round system about the center axle immediately after the child has moved to the location shown in Figure 2? Justify your selection.
(Id+Io)ω0=Isω11, because the sum of the initial angular momenta of the merry-go-round and the child is equal to the final angular momentum of the system.
A disk rotates about its center with an angular speed of 30 rad/s. An identical disk is held at rest above the rotating disk and is then gently dropped on the rotating disk, as shown in Figure 1. The two-disk system then rotates with a common angular speed ω1. A third identical disk is held at rest above the two-disk system. The third disk is gently dropped on the rotating two-disk system, as shown in Figure 2. The three-disk system then rotates with a common angular speed ω2. What is value of ω2?
10 rad/s
A rod may freely rotate about an axis that is perpendicular to the rod and is along the plane of the page. The rod is divided into four sections of equal length of 0.2m each, and four forces are exerted on the rod, as shown in the figure. Frictional forces are considered to be negligible. Which of the following correctly describes an additional torque that must be applied in order to keep the rod from rotating?
18 nm counterclockwise
A rod is initially at rest on a rough horizontal surface. Three forces are exerted on the rod with the magnitudes and directions shown in the figure. The force exerted in the center of the rod is an equidistant 0.5m from both ends of the rod. If friction between the rod and the table prevents the rod from rotating, what is the magnitude of the torque exerted on the rod about its center from frictional forces?
20 nm
A uniform disk with mass M0 and radius R is mounted on a vertical axis so that it can rotate freely in a horizontal plane. The rotational inertia of the disk is I0. The disk is initially at rest. If force F0 is exerted tangentially to the rim of the disk for time interval Δt, the final angular momentum of the disk is L0. Consider a second disk that has the same mass M0 as the first disk, but its radius is 2R. The rotational inertia of this disk is 4I0. If the same force F0 is exerted tangentially to the rim of the disk for the same time interval Δt, then the final angular momentum of the second disk would be
2Linitial
The graph shown represents the net torque that a wrench exerts on a bolt as a function of time as the wrench turns the bolt around its central axis of rotation. What is the change in angular momentum of the bolt after 1000 ms?
3.0 kg⋅m2/s
A point on the edge of a disk rotates around the center of the disk with an initial angular velocity of 3rad/s clockwise. The graph shows the point's angular acceleration as a function of time. The positive direction is considered to be counterclockwise. All frictional forces are considered to be negligible. What is the angular displacement of the point after 10s?
330 radians
A rod is at rest on a horizontal surface. One end of the rod is connected to a pivot that allows the rod to rotate around the pivot after a net external force is exerted on the rod. A lump of clay is launched horizontally toward the free end of the rod, as shown in Figure 1. The lump of clay collides with the rod but does not stick to the rod. The lump of clay comes to rest as the rod rotates around the pivot, as shown in Figure 2. Which of the following linear collisions is analogous to the rotational collision that is described?
A block traveling in the positive direction collides with a second block that is at rest. After the collision, the first block comes to rest and the second block travels at a nonzero speed in the direction that the first object initially traveled.
A rod is at rest on a horizontal surface. One end of the rod is connected to a pivot that allows the rod to rotate around the pivot after a net external force is exerted on the rod. A lump of clay is launched horizontally toward the free end of the rod, as shown in Figure 1. The lump of clay collides with and sticks to the rod, and the clay-rod system rotates, as shown in Figure 2. Which of the following linear collisions is analogous to the rotational collision that is described?
A block traveling in the positive direction collides with a second block that is at rest. After the collision, the two blocks move together with a common final speed.
A group of students must conduct an experiment to determine how the location of an applied force on a classroom door affects the rotational motion of the door. The rotational inertia of the door about its hinges is known. The initial angular velocity of the door is zero. Which of the following lists what measuring devices the students need and the measurements they should take to collect the necessary data to test the relationship between a torque exerted on the door and the change in angular velocity of that object about the hinges of the door? Justify your selection.
A stopwatch to measure the time interval during which the force is applied, a force probe to measure the applied force on the door, a protractor to measure the angular displacement of the door, and a meterstick to measure the radial distance from the door's hinges to the location where the force is applied.
A net torque is applied to the edge of a spinning object as it rotates about its internal axis. The table shows the net torque exerted on the object at different instants in time. How can a student use the data table to determine the change in angular momentum of the object from 0s to 6s? Justify your selection
Create a graph of net torque as a function of time and graph four points of data by using the table. Determine the area bound by the curve and the horizontal axis from 0s to 6s, because the shape of the curve on the graph will be a right triangle and the area can be directly determined.
In an experiment, disk X is held at rest above disk Y that rotates about its center with a constant angular velocity. A student slowly lowers disk X onto disk Y until both disks come into contact and rotate with a common angular velocity. The student collects the necessary data to graph the angular velocity of disk Y as a function of time, as shown in the graph. Both disks are identical. How can the student use the graph to determine the magnitude of the angular impulse on disk Y? Select two answers.
Determine the angular velocity at 1.0s. Determine the angular velocity at 3.0s. Determine the magnitude of the difference of the two angular velocity values. Multiply the result by the rotational inertia of disk Y. Determine the slope of the best-fit curve from 1.75s and 2.25s. Multiply the result by the rotational inertia of disk Y. Multiply the result by 0.5 s.
In an experiment, a solid, uniform disk of mass 0.2kg and radius 0.5m is suspended vertically and can rotate about its center axle such that frictional forces are considered to be negligible. A string is wrapped around the pulley with one end connected to a block of mass 0.1kg that hangs from the string at rest, as shown in Figure 1. The block is released from rest and falls to the ground as the pulley rotates. A student collects the necessary data to construct a graph of the net force exerted on the edge of the pulley as a function of time, as shown in Figure 2. How can the student use the graph in Figure 2 to determine the change in angular momentum of the disk from 0 s to 4 s?
Determine the area bound by the best fit curve and the horizontal axis from 0 s to 4 s and multiply the result by the radius of the disk.
A point on the edge of a disk rotates around the center of the disk with an initial angular velocity of 3rad/s clockwise. The graph shows the point's angular acceleration as a function of time. The positive direction is considered to be counterclockwise. All frictional forces are considered to be negligible.
Determine the area bound by the line and the horizontal axis from 0s to 2s , because this area represents the change in the angular velocity of the point on disk.
A group of students must conduct an experiment to determine how the location of an applied force on a classroom door affects the rotational motion of the door. The rotational inertia of the door about its hinges is known. The initial angular velocity of the door is zero. The students conduct the experiment and gather data about the net torque exerted on the door and angular acceleration of the door for several trials. They create a graph of the door's angular acceleration as a function of the net torque and sketch a best-fit line through the data. How can the students analyze the data to determine the rotational inertia of the door about its hinges?
Determine the inverse of the slope of the best-fit line.
A net force is applied to the edge of a disk that has a diameter of 0.5m. The disk is initially at rest. A graph of the net force as a function of time for the edge of the disk is shown. The net force is applied tangent to the edge of the disk. How can a student use the graph to determine the change in angular momentum of the disk after 8s? Justify your selection.
Determine the vertical intercept, multiply the result by 0.25m, and then multiply that result by 8s. This procedure can be used because ΔL=τΔtΔL=τΔt with τ=rF.
Disk Y of rotational inertia IY about its center is held at rest above disk X of rotational inertia IX about its center. Disk X rotates about its center with an angular velocity +ω1. Disk Y is slowly lowered onto disk X until both disks are in contact and travel together with a common angular velocity. A graph of disk X's angular acceleration α as a function of time is shown. Which of the following equations can a student use to verify that angular momentum is conserved in the situation? Justify your selection.
IXω1=(IX+IY)(ω1−α5t1), because the final velocity of the two-disk system is equal to the initial velocity of disk XX minus the magnitude of the area bound by the curve and the horizontal axis from 32t1to 52t1.
A rigid disk with rotational inertia I0 about its center of mass is initially at rest. The disk is free to rotate without friction about a vertical axis through its center. A constant force F1 that results in a torque τ1 is applied to the edge of the disk. This torque causes the disk to rotate, as shown in Figure 1. After a time Δt0, a second force F2 is also applied to the edge of the disk, resulting in a torque τ2, as shown in Figure 2. Which two of the following predictions are correct about the motion of the disk after F2 is applied to the disk? Select two answers.
If the magnitude τ2=τ1, the disk will continue to spin in the same direction but with zero change in angular velocity per unit time. If the magnitude τ2>2τ1, the disk will eventually come to rest and then spin in the opposite direction with a greater change in angular velocity per unit of time than before τ2τ2 was applied.
Three disks are concentrically attached to one another, and four rods of negligible mass are attached to the outer disk. Identical objects of mass Mo can be attached to the rods, and their positions on the rods can be adjusted. The disks, rods, and objects form a system that freely rotates around a central axis that is perpendicular to the plane of the page. The objects are initially a distance D away from the axis of rotation. A constant force F0 is applied tangent to the second disk, as shown in the figure. How can the system be changed so that the change in angular momentum of the system per unit of time is increased?
Increase the magnitude of the net torque exerted on the system.
An isolated spherical star of radius R0 rotates about an axis that passes through its center with an angular velocity of ω0. Gravitational forces within the star cause the star's radius to collapse and decrease to a value r0<R0, but the mass of the star remains constant. A graph of the star's angular velocity as a function of time as it collapses is shown. Which of the following predictions is correct about the angular momentum L⃗ of the star immediately after the collapse?
L⃗ L→ will be the same as before the collapse.
One end of a horizontal rod is connected to an axle that is connected to a motor that can be adjusted to change the angular speed of the rod as it rotates in a horizontal circle. Students must determine the change in angular momentum of the rod after 10s. The students use the following procedure. Measure the length of the rod with a meterstick. Ensure that the rod is at rest. Start the stopwatch and simultaneously adjust the motor so that the rod rotates around the axle as the angular speed is slowly increased. Record the angular speed of the rod when the stopwatch reads 10s. The students are provided with the equation for the rotational inertia of the rod about one end, I=ML32. Which of the following steps should the students add to the procedure to ensure that the change in angular momentum of the rod can be determined?
Measure the mass of the rod.
In an experiment, a torque of a known magnitude is exerted along the edge of a rotating disk. The disk rotates about its center. All frictional forces are considered to be negligible. Which of the following quantities should a student collect in order to determine the change in angular momentum of the disk for a specific time interval? Justify your selection.
The amount of time the torque is applied to the disk, because the time interval is related to the angular impulse of the disk.
A rod of negligible mass may rotate about a pivot such that frictional forces are considered to be negligible. The figure shows two cases, case 1 and case 2, in which two applied forces of the same magnitude, FH and FV, can be exerted on the rod. Which of the following two statements are correct about the net torque exerted on the rod? Select two answers.
The angular acceleration of the rod in case 1 is greater than the angular acceleration of the rod in case 2. The net torque exerted on the rod in case 2 is zero.
An ice skater rotates in a circle about an internal axis of rotation. Figure 1 shows the skater with her arms extended fully outward. Figure 2 shows the skater with her arms partially inward to her body. Figure 3 shows the skater with her arms completely inward and in contact with her body. Which of the following claims is correct about the angular momentum of the skater?
The angular momentum of the skater is the same in all figures.
A planet of mass Mp orbits a star of mass Ms in the path shown above as a result of Newton's law of universal gravitation. Which of the following predictions correctly describes the angular momentum of the planet-star system and the angular velocity of the planet about the axis of revolution as it travels from position X to position Z?
The angular momentum of the system remains the same, and the angular velocity of the planet increases.
A student conducts an experiment to test the relationship between the net torque exerted on an object and the change in angular momentum of the object. A variable net torque is exerted on the object to make it rotate about its internal axis. Data from the experiment are used to construct a graph of the net torque exerted on the object as a function of time, as shown in Figure 1. A graph is also created of the angular momentum of the object as a function of time, as shown in Figure 2. Which of the following statements about the change in the object's angular momentum for a given time interval is correct? Justify your selection.
The change in the object's angular momentum for a given time interval does not remain the same throughout the experiment. This is because the slope of the best-fit line in Figure 1 is a nonzero constant.
A disk that can freely spin about a central axis is initially at rest until a net force is applied to the disk. The net force is exerted tangentially on the edge of the disk, which has radius 0.5m, mass 0.25kg, and rotational inertia 0.0625kg⋅m2. The magnitude of the force as a function of time is shown in the graph. Which two of the following statements are correct about the disk? Select two answers.
The disk's angular acceleration at 10s is 40rad/s2. The final angular momentum of the disk at 10s is 12.5kg⋅m2/s.
A student is asked to design an experiment to determine the change in angular momentum of a disk that rotates about its center and the product of the average torque applied to the disk and the time interval in which the torque is exerted. A net force is applied tangentially to the surface of the disk. The rotational inertia of the disk about its center is I=12MR2. Which two of the following quantities should the student measure to determine the change in angular momentum of the disk after 10s? Select two answers.
The magnitude of the net force exerted on the disk The distance between the center of the disk and where the net force is applied to the disk
A wooden board of unknown mass is placed on a fulcrum and remains in static equilibrium, as shown in Figure 1. One end of a string is attached to an object of unknown mass, as shown in Figure 2, and can be hung from the board. A student must hang three objects from the wooden board so that the board does not rotate. The student must then mathematically verify whether or not the system remains in static equilibrium. Which two of the following measurements should the student collect in order to make the verification? Justify your selections. Select two answers.
The mass of each object, because each object exerts an external force on the board. The horizontal distance each hanging object is from the tip of the fulcrum, because this distance is perpendicular to the direction of the force that a hanging object applies to the board.
In an experiment, a horizontal cylindrical disk may spin about a central vertical axle. A digital sensor is connected to the axle so that students may measure the angular speed of the disk as it rotates. Students may vary the angular speed of the disk-axle apparatus as data is collected. Students want to plot the magnitude of the angular momentum of the disk as a function of time for a known time interval Δt0. The students are provided with the equation for the rotational inertia of the disk about its center, I=12MR2. What two additional quantities should the students measure or determine in order to produce the desired graph? Select two answers. Justify your selections.
The mass of the disk, because this quantity is necessary to determine the rotational inertia of the disk The radius of the disk, because this quantity is necessary to determine the rotational inertia of the disk
A student conducts an experiment in which data is collected about the net torque exerted on the edge of a disk as it rotates about its center as a function of time. The student creates the graph that is shown. Before the net torque is applied to the disk, it rotates in the positive direction. The student makes the following claim. "The change in angular momentum of the disk from 0 s to 4 s is in the positive direction." 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 area bound by the data and the horizontal axis from 0 s to 4 s is positive.
A student conducts an experiment in which a disk may freely rotate around its center in the absence of frictional forces. 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. "The graph shows that the magnitude of the angular acceleration of the disk decreases 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 disk is constant as time increases.
A group of students must conduct an experiment to determine how the location of an applied force on a classroom door affects the rotational motion of the door. The rotational inertia of the door about its hinges is known. The initial angular velocity of the door is zero. The students must determine how to test the relationship between a torque exerted on the door and the change in the angular velocity of the door. All frictional forces are considered to be negligible. How should the experiment be conducted to test the relationship between the torque exerted on the door and the change in the door's angular velocity in a way that minimizes experimental uncertainty?
The students should perform several trials, applying the same force at different horizontal distances from the hinges.
A uniform rod is at rest on a horizontal surface. A student may launch a sphere of clay toward the rod along one of the three paths shown in the figure. Path X and path Z are directed toward the center of mass of the rod. In each case, the sphere of clay is launched with the same linear speed and sticks to the rod. In each case, the time of collision between the sphere of clay and the rod is time t0. A pivot is fixed to the end of the rod, representing the point at which the rod or clay-rod system may rotate. Frictional forces are considered to be negligible. Consider the situation in which three identical spheres of clay are launched simultaneously, one along each possible path. All three spheres of clay are launched with the same initial linear speed and collide with the rod at the same time. The time of collision with the rod for each sphere is time t0. Which of the following predictions is correct about the motion of the system containing the rod and all three spheres of clay immediately after the collision?
The system will rotate in the clockwise direction with a constant angular speed.
A disk is fixed to a horizontal axle that extends between two supports, as shown in the figure. Frictional forces between the axle and the supports is not negligible. At time ts, the disk rotates about the center axle with an initial angular speed wd. A student measures the angular displacement Δθ0 of a point on the edge of the disk from time ts until the disk no longer rotates. The angular acceleration of the disk is determined to be αd, and this value remains constant. Based on the data, if possible, how could the student predict the angular displacement of a point on the edge of the disk from time ts until the disk no longer rotates if the initial angular speed is increased to 2ωd ? Justify your selection.
Use the equation ω2=ω20+2α(θ−θ0), because the disk comes to rest, ω0=2ωd, and α=αd. Solve for θ−θ0.
In an experiment, one end of a light string is attached and wrapped around a pulley of diameter 0.5 m. The other end of the string is connected to a block of mass 0.5 kg. The block is released from rest, and the pulley begins to spin in the counterclockwise direction, as shown in Figure 1. Students collect the necessary data to create the graph of the magnitude of the angular momentum of the pulley as a function of time shown in Figure 2. The students state that the graph shows that the net torque exerted on the pulley is constant. Do the data from the graph support the students' statement? Justify your selection.
Yes, because the slope of the best-fit line is constant.
During an experiment, students collect data about the angular momentum of a rigid, uniform spinning wheel about an axle as a function of time, which was used to create the graph that is shown. A frictional torque is exerted on the wheel. A student makes the following statement about the data. "The frictional torque exerted on the wheel is independent of the wheel's angular speed." Does the data from the graph support the student's statement? Justify your selection.
Yes, because the slope of the line is constant.
An object of mass M hangs from a string that is looped around a pulley of negligible friction, as shown. The pulley has a mass 0.5M. The object is released from rest and it falls to the floor at time t1. Which of the following pairs of graphs best represents the angular speed as a function of time for the pulley and the vertical speed as a function of time for the falling object for a short time after it is released from rest?
angular: half a trapezoid vertical: triangle
A disk of radius R=0.5cm rests on a flat, horizontal surface such that frictional forces are considered to be negligible. Three forces of unknown magnitude are exerted on the edge of the disk, as shown in the figure. Which of the following lists the essential measuring devices that, when used together, are needed to determine the change in angular momentum of the disk after a known time of 5.0s?
force probe and protractor
A uniform rod is at rest on a horizontal surface. A student may launch a sphere of clay toward the rod along one of the three paths shown in the figure. Path X and path Z are directed toward the center of mass of the rod. In each case, the sphere of clay is launched with the same linear speed and sticks to the rod. In each case, the time of collision between the sphere of clay and the rod is time t0. A pivot is fixed to the end of the rod, representing the point at which the rod or clay-rod system may rotate. Frictional forces are considered to be negligible. A sphere of clay travels toward the rod along path Z. A student must predict what will happen to the linear momentum and the angular momentum of the rod-sphere system as a result of the collision. Which of the following correctly predicts the change, if any, of these quantities?
linear:decreases; angular: no change
A point on a disk rotates around the center axle of the disk. The table shows the angular speed of the point as a function of time. Which of the following graphs could represent the angular position of the point as a function of time?
the one that looks like half of a frown ( my face when I saw the score)
A point on the edge of a disk rotates around the center of the disk with an initial angular velocity of 3rad/s clockwise. The graph shows the point's angular acceleration as a function of time. The positive direction is considered to be counterclockwise. All frictional forces are considered to be negligible. Which of the following graphs qualitatively represents the angular velocity ω of the point on the disk as a function of time t between 0s to 2s?
| / | / |_____/__ | / | /
A point on a rotating object has an initial angular velocity ω0 and rotates with an angular acceleration α0 for a time interval from t=0 to time t=t0. The point then rotates at a constant angular speed until time t=t1. What is the angular displacement of the point from t=0 to t=t1? Express your answer in terms of ω0, α0, t0, t1, and/or any fundamental constants as appropriate.
ω0t0+12α0t20+(ω0+α0t0)t1
A uniform rod is at rest on a horizontal surface. A student may launch a sphere of clay toward the rod along one of the three paths shown in the figure. Path X and path Z are directed toward the center of mass of the rod. In each case, the sphere of clay is launched with the same linear speed and sticks to the rod. In each case, the time of collision between the sphere of clay and the rod is time t0. A pivot is fixed to the end of the rod, representing the point at which the rod or clay-rod system may rotate. Frictional forces are considered to be negligible. Consider the case in which the sphere of clay is launched along path Y. The sphere of clay is launched with a speed v0 and collides with the rod a distance l away from the pivot. The length of the rod is L. The rotational inertia of the rod about the joint is IR, and the mass of the sphere of clay is mc. The sphere of clay is considered to be a point mass. What is the angular speed ωf of the clay-rod system immediately after the collision?
ωf=mcv0l/ mcl2+IR
