physics unit 7 (or 4 )

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?

(MgL cos (θ0))/2

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?

(MgL0)/2

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

1. Angular Velocity vs. Time Graph 2. Angular Position vs. Time Graph

In an experiment, one end of a string is attached to object X, and the other end of the string is attached to object Y. The string is then placed around a pulley, as shown in the figure, and the system containing object X, object Y, the string, and the pulley remains at rest. A support string connects the pulley to the ceiling. Which two of the following measuring devices could students use to mathematically verify that the net torque exerted on the system is zero? Select two answers.

1. Meterstick 2. Electronic Balance

A light string is attached to a massive pulley of known rotational inertia IP, as shown in the figure. A student must determine the relationship between the torque exerted on the pulley and the change in the pulley's angular velocity when the torque is applied for 2.0s. In addition to a stopwatch to measure the time interval, what two measurements could the student make in order to determine the relationship? Select two answers.

1. Radius of the pulley 2. Force exerted on the string to turn the pulley

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.

1. The angular velocity, because this quantity can be determined by calculating the slope of the graph. 2. The translational speed, because v=rω.

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 uniform, rigid rod of length 2m lies on a horizontal surface. One end of the rod can pivot about an axis that is perpendicular to the rod and along the plane of the page. A 10N force is applied to the rod at its midpoint at an angle of 37°. A second force F is applied to the free end of the rod so that the rod remains at rest, as shown in the figure. The magnitude of the torque produced by force F is most nearly

6.0 N * m

A 3kg horizontal disk of radius 0.2m rotates about its center with an angular velocity of 50rad/s. The edge of the horizontal disk is placed in contact with a wall, and the disk comes to rest after 10s. Which of the following situations associated with linear impulse is analogous to the angular impulse that is described?

A 3kg block is initially traveling at 10m/s. The block encounters a 3N frictional force until the block eventually stops.

A solid disk rests on a central axle. A string is wound around the disk, which causes the disk to rotate about its axle when pulled with a constant force FP, as shown in Figure 1. The force is exerted on the string until the string completely unwinds and detaches from the disk at time t=2s. A graph of the net torque as a function of time is shown in Figure 2. How can a student determine what the change in angular momentum of the disk would be between 0s and 2s if friction were negligible? Justify your selection.

The net torque exerted on the disk during the time interval 0 s<t<2 s is 3 N⋅m. After t=2 s the net torque, which is the frictional torque τFriction , is −1 N⋅m . Using this information, it can be inferred that the torque τP due to FP is 4 N⋅m . Therefore, the change in angular momentum from τP , if friction were negligible, would be 4 N⋅m multiplied by 2 s.

One end of a string is attached to the ceiling, and the other end of the string is attached to a cradle that has a meterstick that runs through it. The meterstick can slide through the cradle so that the horizontal position of a point on the meterstick can be changed in the horizontal direction. Students may hang objects of various masses from the meterstick, as shown in the figure. The students notice that when the meterstick-cradle-object-object system is not balanced, the meterstick will rotate. Consider the situation shown above in which the center of the meterstick is aligned with the center of the cradle, which is at a position of x=0m. The system is released from rest. Which of the following claims is correct about the motion of the system containing the meterstick, cradle, and two objects if the system is free to rotate?

The system will rotate in the clockwise direction with an increasing angular speed.

One end of a string is attached to the ceiling, and the other end of the string is attached to a cradle that has a meterstick that runs through it. The meterstick can slide through the cradle so that the horizontal position of a point on the meterstick can be changed in the horizontal direction. Students may hang objects of various masses from the meterstick, as shown in the figure. The students notice that when the meterstick-cradle-object-object system is not balanced, the meterstick will rotate. Consider the situation shown above in which the center of the meterstick is aligned with the center of the cradle, which is at a position of x=0m. Which of the following statements is correct about the torques exerted on the meterstick?

The torque exerted by the 1.0kg object is equal to the torque exerted by the 0.5kg object.

In an experiment, an external torque is applied to the edge of a disk of radius 0.5m such that the edge of the disk speeds up as it continues to rotate. The tangential speed as a function of time is shown for the edge of the disk. The rotational inertia of the disk is 0.125kg⋅m2. Can a student use the graph and the known information to calculate the net torque exerted on the edge of the disk?

Yes, because the change in tangential speed per unit of time can be multiplied by the rotational inertia divided by the radius of the disk.

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

A rod initially at rest as shown in the figure rotates freely around a pivot such that frictional forces are considered to be negligible. The rod is divided equally into four sections, with the length of each segment equal to 2.0m. A downward force F1=100N is applied in the position and direction shown. A student must apply a force F2 to the rod so that the rod remains in equilibrium. To satisfy the condition, what magnitude of force can be exerted on the rod and at what distance away can F2 be applied? Select two answers.

1. Magnitude of F2= 50 & Distance from Pivot = 8 2. Magnitude of F2= 200 & Distance from Pivot = 2

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 [(1/2)*M*(R^2)] about its center of mass. How many forces are applied to the pulley, and how many torques are applied to the pulley about its center when the block is released from rest?

4 Forces and 1 Torque

A cart with a toy projectile launcher attached to its top travels forward at a constant speed v0. The launcher fires a solid sphere forward at a speed much greater than that of the cart-launcher system. The cart's speed after firing the dart is vf<v0. Frictional forces are considered to be negligible. Which of the following scenarios best represents the rotational analog to this situation?

A child on the edge of a large rotating disk throws a ball in the same direction as the child's tangential velocity.

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<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?

Angular Displacement vs. Time Graph beginning at origin, concave up, and increasing

A horizontal disk is at rest on top of an axle, and the friction between the disk and axle is not negligible. In experiment 1, an applied torque is exerted to the edge of the disk for 2s. At that moment, the applied torque is removed, and the disk eventually comes to rest as a result of a frictional torque. Graphs of the disk's angular momentum as a function of time are shown in Figures 1 and 2 for the two experiments. Which of the following statements is correct about the applied torque (tapplied) and frictional torque (tfrictionexerted) on the disk in experiment 1 and experiment 2 ?

Applied Torque : Tapplied 1 < Tapplied 2 Frictional Torque : Tfriction 1 = Tfriction 2

Disk X is held at rest above disk Y, which rotates with angular velocity +ω0 about its center, as shown in the figure. Disk Y is slowly lowered onto disk X until the disks remain in contact and travel together at angular velocity +ω1. Which of the following linear collisions is analogous to the rotational collision that is described?

Block X travels toward block Y with velocity +v1. Block Y is initially at rest. After the collision, block X and block Y travel together with velocity +v2.

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 slope of the line from 0s to 2s, because the slope represents (Δω/Δt).

Two identical disks rotate about their centers in opposite directions with the same magnitude of angular speed ω0. The top disk is dropped onto the bottom disk, as shown in the figure, so they collide and stick together. Which of the following predictions is correct about the motion of each individual disk after the collision?

Each disk will spin with the same final angular velocity ωf where ωf=0.

In an experiment, a solid, uniform cylinder of unknown radius R and unknown rotational inertia I0 about a central axle, is initially at rest. A light string is wrapped around the cylinder, as shown in the figure. The string is pulled with a constant unknown force F0, causing the cylinder to rotate. After an unknown time interval Δt0, the string is completely unwound from the pulley and loses contact with it. To collect data, a student has access to a meterstick, stopwatch, and force probe. The cylinder may rotate but cannot be moved off of its axle. Indicate whether the force applied to the cylinder F0 and the rotational inertia of the cylinder I0 can be measured directly or calculated from measured quantities in the experiment.

F0 - Measured directly I0 - Calculated from measured quantities

Students place the center of mass of a meterstick at the tip of a fulcrum so the meterstick does not rotate. Three objects of various mass are placed at different positions on the meterstick, as shown in the table. A fourth object is placed on the meterstick so that the meterstick and all four objects remain in static equilibrium. Which of the following could represent the mass of the object and its position on the meterstick? Object Mass (kg) Position on Meterstick (m) 1 0.5 20 2 0.3 60 3 0.2 70 4 ? ?

Mass (kg) : 0.50 Position on Meterstick (m) : 66

A uniform cylinder of radius R, mass M, and rotational inertia I0 is initially at rest. The cylinder is mounted so that it is free to rotate with negligible friction about an axle that is oriented through the center of the cylinder and perpendicular to the page. A light string is wrapped around the cylinder. At time t=0, the string is pulled with a constant force F0, which causes the cylinder to rotate, as shown in the figure. After time t=t0, the string is completely unwound from the cylinder and loses contact with it. If necessary, how can a student determine the change in angular momentum ΔL of the cylinder from t=0 to t=t0?

Multiply F0, R, and t0 to calculate the angular impulse. This quantity is equal to ΔL.

The center of mass of a uniform meterstick is placed on a fulcrum. Two objects of known mass, m1 and m2, are hung at known positions on the meterstick. One end of a string is attached to one end of the meterstick, and the other end of the string is looped around a pulley and connected to hanging object X of unknown mass, as shown in the figure. The meterstick does not rotate and is level with the horizontal. Which of the following measuring devices, if any, should be used to make measurements to determine the unknown mass of object X? Justify your selection.

No additional equipment or measurements are needed, because the force due to gravity exerted on objects m1 and m2 and the distance between where the force is applied and the tip of the fulcrum are already known.

In an experiment, a student applies a net force to the edge of a disk for 10s. The student may change the magnitude and the direction of the force at any time in the experiment. A graph of the net torque exerted on the edge of the disk as a function of time is shown. After observing the graph, the student concludes that the angular momentum of the disk has not changed. Does the graph support the student's conclusion? Justify your answer.

No, because the area bound by the curve and the horizontal axis from 0s to 10s is not zero.

A vertically oriented rod may freely rotate around a horizontal axle through its center. A student applies a force on one end of the rod so that the rod rotates around the axis. The student collects the necessary data to generate the graph that is shown. After observing the data, the student makes the following statement. "For the time interval shown, the magnitude of the net torque exerted on the rod decreases over time." Do the data support the student's statement? Justify your selection.

No, because the slope of the best-fit line remains constant.

A uniform disk spins about an axis that passes through the center of the disk and is perpendicular to the plane of the disk, as shown in Figure 1. The disk has an initial angular velocity of ωd and uniformly accelerates to rest over time. The angular velocity of the disk as a function of time is shown in Figure 2. A student must determine the angular displacement of a point on the edge of the disk from t=0 to the instant in time the disk comes to rest if the point's initial velocity is changed to 2ωdbut its angular acceleration is the same as shown in Figure 2. How can the graph in Figure 2 be changed before the student can determine the angular displacement? Justify your selection.

Recreate the graph with a vertical intercept that is twice the value of the intercept shown in Figure 2, because the angular velocity is increased from ωd to 2ωd. The slope of the line should be the same in both graphs, because the angular acceleration is the same in both graphs.

Two students, X and Y, sit on a seesaw that is in static equilibrium, as shown in the figure. A fulcrum is located at the center of the seesaw. The mass mY of Student Y is known. A third student, Z, must determine the mass of Student X. Which of the following measurements should Student Z make in order to determine the mass of Student X? Justify your selection.

The distance that Student X is from the center of the seesaw and the distance that Student Y is from the center of the seesaw, because these are the distances from the fulcrum to the location of the forces exerted on the two-student-seesaw system.

In an experiment, students must determine the angular momentum of a vertically rotating nonuniform disk of diameter 1.0m. A central axle of negligible friction is located at the center of the disk and is oriented perpendicular to the plane of the page. A string is wound around the disk, and an object of unknown mass is attached to the string, as shown in Figure 1. The object is released from rest, which causes the disk to rotate as the object falls. The disk's angular position as a function of time is shown in Figure 2. The tension in the string is 5N and remains constant. Which of the following mathematical routines could a student use to determine the change in angular momentum of the disk during a time interval Δt ? Justify your selection.

Use ΔL=τΔt, where τ=rFsinθ with r=0.5m, F=5N, and θ=90 degrees.

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.

A group of students conduct an experiment on the rotational motion of a horizontal, rigid rod that is connected to a motor, as shown in the figure. The students collect the data necessary to create a graph of the angular velocity of the disk as a function of time. One student in the group states that the graph shows that a net external torque is exerted on the rod, because the rod's angular momentum increased. Do the data support the student's statement? Justify your selection.

Yes, because the angular velocity increases with time. Therefore, the angular momentum of the rod increases with time. This can only occur as the result of a net torque.

In an experiment, a disk of known rotational inertia Irotates as a result of a known force that is applied to the disk's edge. The angular velocity of the disk is measured as a function of time, as shown in the graph. Students must determine the relationship between the magnitude of the change in angular momentum of the disk from t=0s to t=10s and the torque applied to the disk. What quantity could the students measure in order to make the determination? Justify your selection.

the radius r of the disk, because τ=Fr sin θ

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 [(1/2)*M*(R^2)] about its center of mass. Which of the following graphs, if any, shows the angular velocity ω of the pulley as a function of time t after the block is released from rest

w vs. t graph, begins at origin, line increasing with a slope of 1

Two identical wheels, wheel 1 and wheel 2, initially at rest begin to rotate with constant angular accelerations α. After rotating through the same angular displacement, Δθ0 , the angular velocity of wheel 1 is ω1 and the angular velocity of wheel 2 is ω2=3ω1 . How does the angular acceleration of wheel 2, α2, compare to the angular acceleration of wheel 1, α1 ?

α2=9α1