CH08 Physics

The rod of length a is free to rotate about an axis through the center. Five forces of equal magnitude F are acting on the rod as shown. What is the net torque on the rod about the axle?

0 Reason: The upward forces on the left and the downward forces on the right result in torques that are clockwise, while the downward forces on the left and the upward forces on the right result in torques that are counterclockwise.

A light rod of length L is free to rotate about an axis passing perpendicularly through its center. Two masses m are hanging at the positions shown in the figure. What is the magnitude of the angular acceleration of the system in rad/s2 immediately after the rod is released from rest?

g/5a

Rank the following with the largest torque magnitude at the top and the smallest at the bottom. The torque is exerted by the force about an axis passing perpendicularly through the point marked with a dot in the diagrams. All of the forces have the same magnitude.

C D A B

Four rods of mass m and length a are free to rotate about one end. The directions of the forces change when the rods rotate such that they remain perpendicular to the rod at all times. Rank the work done by the torque in each of the cases, with the largest work at the top and the smallest at the bottom.

D C A B

The dumbbell shown consists of two masses m and 2m connected by a light plastic rod. It is free to rotate at angular speed ω about the axes perpendicular to the plane of the dumbbell at the five points A through E shown (denoted by the × mark). Rank the points according to the rotational kinetic energy of the dumbbell about the five points, from largest at the top to smallest at the bottom.

E D A C B

During Hurricane Katrina, a news reporter standing still on flat, level ground near New Orleans was able to lean forward without any support into the wind at an angle of 30 degrees with respect to the vertical without falling over. Which forces acting on the news reporter were distributed rather than acting at a point?

Force of the air caused by the wind Force of gravity caused by the Earth Normal force caused by the ground Force of static friction caused by the ground

A ballet dancer leans forward with her arms extended forward and her left leg extended backward, and balances on the toes of her right foot. Her body is in equilibrium. Select all of the following locations through which the axis of rotation can pass when calculating the torques exerted on her body.

Her center of gravity Her center of mass Where her toes touch the floor

A hoop and a solid disk of the same radius and mass are given the same initial speed at the bottom of an inclined plane. The objects roll up the incline without slipping. Which object will move the farthest distance up the incline before rolling back down the incline?

Hoop

If a child is spinning in an office chair and pulls her arms and legs in so that her moment of inertia is reduced by half, what happens to her angular speed?

It doubles.

A wheel is spinning counterclockwise and spinning faster. Only one torque is acting on the wheel. Is the torque positive, negative, or zero?

Positive

Select all of the following that are true regarding the work done from the torque.

The work is directly proportional to the magnitude of the angular displacement. The work is directly proportional to the magnitude of the torque. The work can be positive, negative, or zero.

A car at rest begins to move away from you and continues to speed up. What is the direction of the net torque acting on the wheels?

To the left

A car driving away from you applies its brakes so that it begins to slow down. What is the direction of the net torque applied to the wheels?

To the right

You are watching a car moving directly toward you. The car's tires are rolling without slipping. What is the direction of the angular momentum of the car's tires?

To your right

Match the condition on the left with the type of equilibrium on the right.

Zero net force matches Translational equilibrium Zero net torque matches Rotational equilibrium

Objects A and B having equal moments of inertia are rotating about the same axis of rotation. The rotational kinetic energy of A is twice the rotational kinetic energy of B. Which of the following is true regarding the angular speeds of A and B?

ωA = √2 ωB

A uniform thin rod of mass M and length L rotates about an axle through its center of mass at an angular acceleration due to a force applied at one end of the rod. If the length of the rod and the magnitude of the applied force were both doubled and the force still acted at one end of the rod, then by what factor would the magnitude of the angular acceleration change? Assume that the mass of the rod is constant.

1

The axis of rotation of an object passes perpendicularly through its center. What is the approximate length of the lever arm if a force is applied at a location 4 meters directly above the center of the object and in a direction up and to the left at a 45-degree angle above the horizontal?

2.83 m

Which of the following would make the rotational kinetic energy of a particle moving in a circle double?

Double the mass

Pick the option below that would leave the magnitude of the angular momentum unchanged of an object with a moment of inertia I and angular speed ω.

Double the moment of inertia and cut the angular speed in half Cut the moment of inertia in half and double the angular speed

The structure shown is constructed from three identical pieces of metal. Which of the following rankings of the moments of inertia of the structure about the four axes A through D is correct?

IA = ID > IC > IB

Under which circumstances is an object's center of gravity equal to its center of mass?

Only when the gravitational field is uniform in both magnitude and direction

When considering the force of __________ acting on an extended object, we can model the force as though it were only acting on a single point called the center of gravity.

gravity

Conservation of angular momentum applies to both the very big and the very small. When an electron drops to a lower orbit, a photon is created. If the photon has one unit of angular momentum, then the electron's angular momentum must change by _____________ unit.

one

When an object rolls without slipping, then the center of mass acceleration is equal to the product of the angular acceleration and the _______

radius

For the seesaw in the figure, assume that both masses are the same. At what distance x does the second mass need to be placed for the seesaw to be in rotational equilibrium?

x = 1.0 m

A uniform beam of mass 10 kg has a length of 10 meters. It is oriented horizontally, that is, it is level, not tilted. Its left end is located at x = 0 m, and its right end is located at x = 10 m. A narrow support is located under the beam at x = 4 m. Where must a narrow 10 kg object be placed on top of the beam so that the beam is in rotational equilibrium?

x = 3 m Reason: The beam's center of mass is located at x = 5 m.

Objects A and B having equal moments of inertia are rotating about the same axis of rotation. The rotational kinetic energy of A is one half of the rotational kinetic energy of B. Which of the following is true regarding the angular speeds of A and B?

ωA = ωB/√2

Select all of the following conditions that are true when an object is in equilibrium (both translational and rotational).

The net force is zero. The net torque is zero. The (translational) acceleration is zero.

Two identical objects are both rotated through the same number of rotations with the same torque, but the first was done in half the time. Choose the true statement from below.

The same work was done on each but the power for the first was twice that of the second.

A 1.0 m long rod is free to rotate at O. What is the distance d such that the torque exerted by force F2−→F2→ has the same magnitude as the torque due to force F1−→F1→, given that F1 = F2/4?

0.50 m

A system consists of three small objects of equal mass that are connected to a common central point by rigid rods of negligible mass. The axis of rotation for all three objects passes through the central point. The lengths of the three rigid rods are given below. Rank the moment of inertia of the system, with the largest moment of inertia at the top and the smallest at the bottom. Instructions Choice 1 of 4. l1 = 1 cm, l2 = 2 cm, l3 = 7 cm Choice 2 of 4. l1 = 1 cm, l2 = 2 cm, l3 = 10 cm Choice 3 of 4. l1 = 4 cm, l2 = 4 cm, l3 = 4 cm Choice 4 of 4. l1 = 5 cm, l2 = 5 cm, l3 = 5 cm

1, 2, 10 5, 5, 5 1, 2, 7 4, 4, 4

Two masses are placed on a seesaw. The mass on the left weighs 10 N. What is the weight of the mass on the right if the seesaw is in rotational equilibrium?

25 N

Rank the following situations according to the mechanical advantage, with the largest mechanical advantage at the top and the smallest at the bottom.

1. A person jacks a 10-000N car using 200N of force 2. a 2500N piano using 250N of force 3. a 4000N of gravel in wheelbarrow using 1000N of force

Rank the following situations according to the mechanical advantage, with the largest mechanical advantage at the top and the smallest at the bottom.

1. a person pried the sewer lid 2. a person lifts a statue in a wheel barrow 3. a person cracks a walnut with pliers

Rank the following situations involving a solid sphere free to rotate about a line passing through its center according to the magnitude of the net torque, with the largest net torque magnitude at the top and the smallest at the bottom. Also worded as: Rank the following situations involving a solid sphere free to rotate about a line passing through its center according to the magnitude of its angular momentum, with the largest angular momentum magnitude at the top and the smallest at the bottom.

1. m = m, r = 3r, ang. accel. mag. = a 2. m = 2m, r = 2r, ang. accel. mag. = a 3. m = 2m, r = r, ang. accel. mag. = 2a 4. m = 3m, r = r, ang. accel. mag. = a

Rank the following situations according to the mechanical advantage, with the largest mechanical advantage at the top and the smallest at the bottom.

1. pushes a tire jack handle down 20 cm to move a car up 4 mm 2. pulls a rope down 10 m to move a piano up 1 m 3. rotates a screwdriver around a circle of circum. 8 cm to rotate a screw around a ... 2 cm

Rank the following objects, each rolling without slipping down a separate ramp inclined at a different angle, according to the magnitude of the acceleration of each of their centers of mass, with the largest magnitude of the acceleration of the center of mass at the top and the smallest at the bottom.

1. solid sphere r=10cm accelerating at 10 rad/s2 2. hollow sphere r=2cm accel=40rad/s2 3. solid cylinder r=10cm accel=5 rad/s2 4. hollow cylinder r=10cm accel=4 rad/s2

The axis of rotation of an object passes perpendicularly through its center. What is the approximate length of the lever arm if a force is applied at a location 2 meters directly below the center of the object and in a direction down and to the right at a 45-degree angle below the horizontal?

1.41 m

A particle of mass m is rotating about a fixed point at a distance r. The angular speed of the particle is ω. What is the rotational kinetic energy of the particle?

1/2mr2ω2

In a clutch-flywheel system used in automobiles, two disks initially separate are brought together in contact. Before being brought together, disk A is at rest and disk B is spinning at 3000 rpm. After being brought together, both disks rotate at 1000 rpm. What is the moment of inertia of disk A divided by the moment of inertia of disk B?

2

An unpowered pottery wheel is rotating at 30 rpm. If a blob of clay is dropped straight down on the wheel, increasing the moment of inertia of the system by 50%, what is the angular speed of the wheel after the clay lands on it.

20 rpm

A uniform log is rolling down hill without slipping and picking up speed. If the center of mass acceleration is 3.12 m/s2, and the radius is 15 cm, what is the angular acceleration?

20.8 rad/s2

The triangular structure shown consists of two masses connected to each other and to the origin via three light plastic rods. The system is rotating about a line passing perpendicularly through the point O at a constant angular speed ω. What is the rotational kinetic energy of the system?

6(1/2ma2ω2)

A constant torque of 1.00 N⋅m is applied to a wheel that starts from rest and makes one revolution in one second. What was the average power due to this torque during this time?

6.28 W

A wheel of radius 10 cm rolls without slipping along a flat, horizontal surface for 10 seconds at constant angular speed. During this time, it makes 10 complete revolutions. How far did its center of mass move during this time?

6.28 m

Planet X orbits a star in the Andromeda galaxy. When planet X is closest to the star, the distance between the center of the planet and the center of the star is 1.5 x 1011 m and the planet's orbital speed is 50,000 m/s. What is the distance between their centers when the planet's orbital speed is 10,000 m/s? Assume that the star is stationary at all times.

7.5 x 10^11 m

An electron is moving with uniform circular motion around a circle of radius 6 cm. Its initial rotational kinetic energy is 24 J. If its (translational) speed doubled and it kept moving around the same circle, then how much rotational kinetic energy would it have?

96 J

A model of a rectangular molecule is constructed out of particles of equal mass connected by light rods as shown. Rank the moment of inertia of the molecule about the axes A, B, and C, with the largest moment of inertia at the top and the smallest at the bottom.

A C B

The structure shown below consists of three particles placed at the three corners of an isosceles right triangle connected by light rods. Rank the locations A, B, and C according to the moment of inertia of the structure about an axis going perpendicularly through each point, with the largest moment of inertia at the top and the smallest at the bottom.

A C B

A uniform shelf of known length and mass is suspended from two wires, and has three small statues of known masses at known locations sitting on it. Which locations would it be most helpful to pick as the axis of rotation if we want to solve for the tensions?

At the locations where wire makes contact with the shelf. Reason: If the point is chosen so that one of the forces produces no torque, that force will cancel out. Use this to eliminate one of the unknown forces.

A cube rests on a flat surface. If the flat surface is inclined at an angle of 45 degrees with respect to the horizontal, then where is the point of application of the contact forces between the cube and the flat surface?

At the lower corner of the cube that is in contact with the flat surface

A cube rests on a flat surface. If the flat surface is horizontal, then where is the point of application of the contact forces between the cube and the flat surface?

At the middle of the bottom of the cube that is in contact with the flat surface

A wheel is free to rotate about a line passing perpendicularly through its center. Only one torque is applied to the wheel. From your point of view (looking along the axis of rotation), the torque causes the wheel to rotate faster in the clockwise direction. What is the direction of the torque exerted on the wheel?

Away from you

You are watching the second hand of a clock go around the face smoothly without pausing or jerking. What is the direction of the angular momentum of the second hand?

Away from you

In each of the following situations, a pulley in the shape of a disk is free to rotate about a fixed axle through the center. A light rope is wrapped around the pulley, and a mass is attached to the loose end of the rope. All three pulleys are released from rest. Rank them according to the angular acceleration, from the largest angular acceleration magnitude at the top to the smallest at the bottom.

C A B

A turntable rotates at a constant angular velocity. Three pennies glued to the surface at three different distances from the center rotate with the turntable. Rank the following according to the moment of inertia of the pennies, with the largest moment of inertia at the top and the smallest at the bottom.

C B A

A Styrofoam ball and a basketball have the same mass and same diameter. Both are rotating about a line passing through their centers at the same angular velocity. Which of the following is true? Assume that the basketball is hollow and that the Styrofoam ball is solid.

IStyrofoam ball < Ibasketball

Select all of the following statements that are always true.

If every force is zero, then the every torque is zero as well. It is impossible to exert a nonzero torque without exerting a nonzero force.

A disk is rotating freely about an axis that passes perpendicularly through its center at a constant angular speed. Two ants of equal mass suddenly drop onto the edge of the disk and get stuck to the rotating disk. What happens to the angular speed of the rotating disk?

It decreases, but not to zero.

A small object is connected to a rigid rod of negligible mass. The small object is free to rotate about a line that passes perpendicularly through the other end of the rigid rod. If the mass of the small object doubles and the length of the rigid rod stays the same, then what happens to the moment of inertia of the small object?

It doubles.

The figure shows a disk with two ants of equal mass standing at the edge. The disk is rotating freely about an axis that passes perpendicularly through its center at a constant angular speed. The two ants start walking toward the center of the disk at a constant speed. What happens to the angular speed of the disk?

It increases.

Most objects in orbit travel in elliptical orbits, so that their distance to the object they are orbiting is constantly changing. Match how each quantity changes when the orbiting object is closest to the object being orbited. Instructions

Orbital speed matches Greatest Moment of inertia matches Least Angular Momentum matches Unchanged

A wheel is spinning clockwise and spinning slower. Only one torque is acting on the wheel. Is the torque positive, negative, or zero?

Positive Reason: Since the wheel is spinning clockwise and spinning slower, its angular acceleration and the torque acting on it are counterclockwise, which is positive.

A particle of mass m subject to a force rotates in a circle of radius r with an angular acceleration. Select all of the following that will decrease the magnitude of the angular acceleration by a factor of 4.

Quadruple r Quadruple m

Select all of the following that are true regarding the average power due to a constant torque.

The average power is directly proportional to the magnitude of the torque. The average power is directly proportional to the magnitude of the average angular speed. The average power can be positive, negative, or zero.

Select all of the following statements that are true regarding how to choose the axis of rotation in equilibrium problems.

The axis of rotation can be chosen arbitrarily. The best place to choose the axis is usually at the point of application of an unknown force.

A solid disk, a hoop, and a block of ice, each of the same mass, are one at a time placed at the same spot on an incline and released. The disk and hoop roll without slipping, but treat the block of ice as though it were frictionless. Choose all the true statements.

The block has the greatest center of mass speed at the bottom of the incline. Each has the same kinetic energy at the bottom of the incline.

A heavy cabinet is being pulled up a ramp at constant speed. There is friction between the bottom of the cabinet and the ramp. One end of a rope is attached to a handle on the side of the cabinet, and the other end is pulled by a person standing at the top of the ramp. Which forces acting on the cabinet and the handle are distributed rather than acting at a point? Treat the cabinet and the handle as a single object. Ignore air resistance.

The force of gravity caused by the Earth The normal force caused by the ramp The force of kinetic friction caused by the ramp

Select all of the following that are needed to determine the torque produced by a force.

The location of the axis of rotation The location where the force is applied to the object The direction and magnitude of the force

Select all of the following combinations that are possible. Assume that there is only one torque acting on the object or system of particles.

Torque > 0 (counterclockwise), angular velocity < 0 (clockwise), angular acceleration > 0 (counterclockwise) Torque < 0 (clockwise), angular velocity > 0 (counterclockwise), angular acceleration < 0 (clockwise) Reason: If there is only one torque, then the torque and angular acceleration are always in the same direction.

Select all of the following statements that are true regarding torque.

Torque can be positive, negative, or zero. The SI units of torque are N⋅m.

Select all of the following statements that are true regarding torque.

Torques can cause objects to rotate faster or slower. It is impossible to exert a torque without exerting a force.

A wheel is free to rotate about a line passing perpendicularly through its center. Only one torque is applied to the wheel. From your point of view (looking along the axis of rotation), the torque causes the wheel to rotate faster in the counterclockwise direction. What is the direction of the torque exerted on the wheel?

Toward you

The apparatus shown below is used to study rotational dynamics in a physics lab. The wheel is a solid uniform disk of mass M and radius R, and is free to rotate about a horizontal axle going through its center. The light rope that is wrapped around the pulley is pulled down with a force F. What is the angular acceleration of the disk?

α = 2F/MR