Question 1

The fundamental dimensions of angular momentum are: A) mass·length·time^-¹ B) mass·length^-²·time^-² C) mass·²·time^-¹ D) mass·length²·time^-² E) none of these

Accepted Answer

E

Question 2

Possible units of angular momentum are: A) kg.m/s B) kg.m²/s² C) kg.m/s² D) kg.m²/s E) none of these

Accepted Answer

Angular momentum is a product of linear momentum and the radius of the object's path, commonly denoted as L = r x p, where r is the radius vector and p is the linear momentum. The unit of linear momentum is kg m/s, and the unit of radius is m. Multiplying these units gives kg m^2/s. Therefore, the unit of angular momentum is kg m^2/s. Choice D is the only option with this unit, and it also includes the appropriate symbols for force (F), length (m), and time (s) in the correct order of dimensions.

Question 3

A cylinder of radius R = 6.0 cm is on a rough horizontal surface. The coefficient of kinetic friction between the cylinder and the surface is 0.30 and the rotational inertia for rotation about the axis is given by MR²/2, where M is its mass. Initially it is not rotating but its center of mass has a speed of 7.0 m/s. After 2.0 s the speed of its center of mass and its angular velocity about its center of mass, respectively, are: A) 1.1 m/s, 0 B) 1.1 m/s, 19 rad/s C) 1.1 m/s, 98 rad/s D) 1.1 m/s, 200 rad/s E) 5.9 m/s, 98 rad/s

Accepted Answer

D

Question 4

A solid wheel with mass M, radius R, and rotational inertia MR²/2, rolls without sliding on a horizontial surface. A horizontal force F is applied to the axle and the center of mass has an acceleration a. The magnitudes of the applied force F and the frictional force f of the surface, respectively, are: A) F = Ma, f = 0 B) F = Ma, f = Ma/2 C) F = 2Ma, f = Ma D) F = 2Ma, f = Ma/2 E) F = 3Ma/2, f = Ma/2

Accepted Answer

The answer of A solid wheel with mass M, radius...

Question 5

A wheel of radius 0.5 m rolls without sliding on a horizontal surface as shown. Starting from rest, the wheel moves with constant angular acceleration 6 rad/s². The distance in traveled by the center of the wheel from t = 0 to t = 3 s is: A) zero B) 27 m C) 13.5 m D) 18 m E) none of these

Accepted Answer

Let's first find the angular velocity at t=3s using the formula: 
ω = ω0 + αt 
where ω0 = 0 (starting from rest), α = 6 rad/s^2, and t = 3s 
ω = 0 + 6(3) = 18 rad/s

Now, let's find the distance traveled by the center of the wheel using the formula: 
s = rθ 
where r = 0.5m (radius of the wheel) and θ is the angle rotated by the wheel in radians

To find θ, we can use the formula: 
θ = ω0t + 0.5αt^2 
where ω0 = 0 (starting from rest), α = 6 rad/s^2, and t = 3s 
θ = 0 + 0.5(6)(3)^2 = 27 rad

Therefore, the distance traveled by the center of the wheel is: 
s = rθ = (0.5)(27) = 13.5 m

So, the best choice is C.

Question 6

The coefficient of static friction between a certain cylinder and a horizontal floor is 0.40. If the rotational inertia of the cylinder about its symmetry axis is given by I = (1/2)MR², then the maximum acceleration the cylinder can have without sliding is: A) 0.1 g B) 0.2 g C) 0.4 g D) 0.8 g E) g

Accepted Answer

The answer of The coefficient of static friction between a...

Question 7

A 5.0-kg ball rolls without sliding from rest down an inclined plane. A 4.0-kg block, mounted on roller bearings totaling 100 g, rolls from rest down the same plane. At the bottom, the block has:
A) greater speed than the ball
B) less speed than the ball
C) the same speed as the ball
D) greater or less speed than the ball, depending on the angle of inclination
E) greater or less speed than the ball, depending on the radius of the ball

Accepted Answer

We can use conservation of energy to solve this problem. The potential energy at the top of the incline is converted into kinetic energy at the bottom.

For the ball:

- Potential energy at the top = mgh
- Kinetic energy at the bottom = 1/2mv^2
- Since there is no friction, all the potential energy is converted into kinetic energy: 
mgh = 1/2mv^2
- Solving for v: 
v = sqrt(2gh)

For the block:

- The block has rotational as well as translational kinetic energy, so we need to use the parallel axis theorem to find the moment of inertia of the block-roller system: 
I = (1/2)MR^2 + M(r+R)^2 - Mr^2
(where M is the mass of the block and r and R are the radii of the rollers and the block respectively)
- The total mass of the block-roller system is 4.1 kg (4 kg for the block and 0.1 kg for the rollers), so the force of gravity acting on the system is:
F = (4.1 kg)(9.81 m/s^2) = 40.221 N
- The net torque on the system is:
τ = Frsinθ
(where θ is the angle of inclination and rsinθ is the effective radius of the rollers)
- Since the block-roller system starts from rest, its initial angular velocity is zero. Applying the rotational version of conservation of energy: 
mgh = 1/2mv^2 + 1/2Iω^2
(where ω is the final angular velocity)
- Substituting in for I and τ: 
mgh = 1/2mv^2 + 1/2[(1/2)MR^2 + M(r+R)^2 - Mr^2] (v/rsinθ)^2
- Solving for v: 
v = 2rsinθ sqrt[gh/(5rsinθ + 2R + 2r/3)]
- Note that the velocity of the block depends on the angle of inclination as well as the radii of the block and rollers. However, we can see that the velocity of the ball is independent of these factors:
v(ball) = sqrt(2gh)
- Since g, h, and the masses of the ball and block are the same, we can compare the velocities by comparing the square roots: 
sqrt[2gh] > 2rsinθ sqrt[gh/(5rsinθ + 2R + 2r/3)]
- The right-hand side of this inequality is always less than 1, so we have: 
sqrt[2gh] > 2rsinθ sqrt[gh/(5rsinθ + 2R + 2r/3)] > 1
- Therefore, the ball has a greater speed than the block. Answer choice A is correct.

Question 8

A yo-yo, arranged as shown, rests on a frictionless surface. When a force  is applied to the string as shown, the yo-yo: 
A) moves to the left and rotates counterclockwise
B) moves to the right and rotates counterclockwise
C) moves to the left and rotates clockwise
D) moves to the right and rotates clockwise
E) moves to the right and does not rotate

Accepted Answer

According to Newton's third law, there will be an equal and opposite force on the yo-yo by the string (11eaafba_af3c_b2d5_a7a5_431c0585b0c6_TB3976_11). Therefore, the yo-yo will move to the right (in the direction of the force) with the same acceleration as that of the force.

Now, since the force is applied at a distance from the center of mass, the yo-yo will also experience a torque, causing it to rotate counterclockwise. This is because the force tries to move the yo-yo in a straight line to the right, but the string restricts the motion by pulling upwards. As a result, the yo-yo experiences a torque directed towards the left, causing it to rotate counterclockwise. Therefore, the correct answer is option B, i.e., the yo-yo moves to the right and rotates counterclockwise.

Question 9

When the speed of a rear-drive car is increasing on a horizontal road the direction of the frictional force on the tires is:
A) forward for all tires
B) backward for all tires
C) forward for the front tires and backward for the rear tires
D) backward for the front tires and forward for the rear tires
E) zero

Accepted Answer

When a rear-drive car accelerates, the torque causes the rear wheels to rotate faster than the front wheels. This creates a force that pushes the car forward, but also creates a force that tries to rotate the car around its center of gravity. The result is that the front tires experience a backwards frictional force, and the rear tires experience a forwards frictional force. Therefore, the correct answer is D - backward for the front tires and forward for the rear tires.

Question 10

A wheel rolls without slipping along a horizontal road as shown. The velocity of the center of the wheel is represented by $\rightarrow$. Point P is painted on the rim of the wheel. The instantaneous velocity of point P is: 
A) $\rightarrow$
B)$\leftarrow$
C) $\uparrow$
D) 
E) zero

Accepted Answer

The answer of A wheel rolls without slipping along a...

Question 11

A forward force acting on the axle accelerates a rolling wheel on a horizontal surface. If the wheel does not slide the frictional force of the surface on the wheel is:
A) zero
B) in the forward direction and does zero work on the wheel
C) in the forward direction and does positive work on the wheel
D) in the backward direction and does zero work on the wheel
E) in the backward direction and does positive work on the wheel

Accepted Answer

The answer of A forward force acting on the axle...

Question 12

A single force acts on a particle situated on the positive x axis. The torque about the origin is in the negative z direction. The force might be:
A) in the positive y direction
B) in the negative y direction
C) in the positive x direction
D) in the negative x direction
E) in the positive z direction

Accepted Answer

If the torque is in the negative z direction, then the force must be in the negative y direction (since the origin is on the positive x-axis). This is because the torque is given by the cross product of the position vector and the force vector: 
$\bf{	au} = \bf{r} 	imes \bf{F}$ 
where $\bf{	au}$ is the torque, $\bf{r}$ is the position vector, and $\bf{F}$ is the force vector. Since the position vector points in the positive x direction and the torque is in the negative z direction, the force vector must point in the negative y direction to produce that torque.

Question 13

Two uniform cylinders have different masses and different rotational inertias. They simultaneously start from rest at the top of an inclined plane and roll without sliding down the plane. The cylinder that gets to the bottom first is:
A) the one with the larger mass
B) the one with the smaller mass
C) the one with the larger rotational inertia
D) the one with the smaller rotational inertia
E) neither (they arrive together)

Accepted Answer

For rolling objects without slipping, the acceleration down an incline does not depend on mass or rotational inertia but on the shape (which determines the distribution of mass) of the object. Since both objects are uniform cylinders, they have the same shape factor in their moment of inertia calculation, leading to the same acceleration and thus, they reach the bottom at the same time regardless of their mass or rotational inertia differences.

Question 14

Two identical disks, with rotational inertia I (= 1/2 MR²), roll without slipping across a horizontal floor and then up inclines. Disk A rolls up its incline without sliding. On the other hand, disk B rolls up a frictionless incline. Otherwise the inclines are identical. Disk A reaches a height 12 cm above the floor before rolling down again. Disk B reaches a height above the floor of: A) 24 cm B) 18 cm C) 12 cm D) 8 cm E) 6 cm

Accepted Answer

The answer of Two identical disks, with rotational inertia I...

Question 15

When we apply the energy conversation principle to a cylinder rolling down an incline without sliding, we exclude the work done by friction because:
A) there is no friction present
B) the angular velocity of the center of mass about the point of contact is zero
C) the coefficient of kinetic friction is zero
D) the linear velocity of the point of contact (relative to the inclined surface) is zero
E) the coefficient of static and kinetic friction are equal

Accepted Answer

The energy conservation principle can be applied without considering the work done by friction because the point of contact between the cylinder and the incline does not move relative to the inclined surface (i.e., its linear velocity is zero), indicating that there is no work done by friction in the direction of motion. This is characteristic of pure rolling motion, where static friction is present but does not do work because there is no relative sliding motion at the point of contact.

Question 16

A thin-walled hollow tube rolls without sliding along the floor. The ratio of its translational kinetic energy to its rotational kinetic energy (about an axis through its center of mass) is:
A) 1
B) 2
C) 3
D) 1/2
E) 1/3

Accepted Answer

The answer of A thin-walled hollow tube rolls without sliding...

Question 17

A sphere and a cylinder of equal mass and radius are simultaneously released from rest on the same inclined plane sliding down the incline. Then:
A) the sphere reaches the bottom first because it has the greater inertia
B) the cylinder reaches the bottom first because it picks up more rotational energy
C) the sphere reaches the bottom first because it picks up more rotational energy
D) they reach the bottom together
E) none of the above is true

Accepted Answer

The answer of A sphere and a cylinder of equal...

Question 18

Two wheels roll side-by-side without sliding, at the same speed. The radius of wheel 2 is twice the radius of wheel 1. The angular velocity of wheel 2 is:
A) twice the angular velocity of wheel 1
B) the same as the angular velocity of wheel 1
C) half the angular velocity of wheel 1
D) more than twice the angular velocity of wheel 1
E) less than half the angular velocity of wheel 1

Accepted Answer

Since both wheels roll without sliding at the same speed, the wheel with the larger radius (wheel 2) will have a lower angular velocity. This is because angular velocity ($\omega$) is related to linear velocity ($v$) and radius ($r$) by the equation $\omega = \frac{v}{r}$. Therefore, if the radius of wheel 2 is twice that of wheel 1, its angular velocity will be half.

Question 19

A hoop, a uniform disk, and a uniform sphere, all with the same mass and outer radius, start with the same speed and roll without sliding up identical inclines. Rank the objects according to how high they go, least to greatest.
A) hoop, disk, sphere
B) disk, hoop, sphere
C) sphere, hoop, disk
D) sphere, disk, hoop'
E) hoop, sphere, disk

Accepted Answer

The height each object reaches on the incline depends on its rotational inertia. A hoop has the largest rotational inertia, followed by a disk, and then a sphere has the smallest. Objects with larger rotational inertia convert more of their kinetic energy into rotational motion, leaving less energy for potential energy at a height. Therefore, the hoop goes the least high, the disk goes higher, and the sphere goes the highest.

Question 20

A hoop rolls with constant velocity and without sliding along level ground. Its ratation kinetic energy is:
A) half its translational kinetic energy
B) the same as its translational kinetic energy
C) twice its translational kinetic energy
D) four times its translational kinetic energy
E) one-third its translational kinetic energy

Accepted Answer

The answer of A hoop rolls with constant velocity and...

Quiz 11: Rolling, Torque, and Angular Momentum

The fundamental dimensions of angular momentum are:

Possible units of angular momentum are:

A wheel of radius 0.5 m rolls without sliding on a horizontal surface as shown. Starting from rest, the wheel moves with constant angular acceleration 6 rad/s². The distance in traveled by the center of the wheel from t = 0 to t = 3 s is:

The coefficient of static friction between a certain cylinder and a horizontal floor is 0.40. If the rotational inertia of the cylinder about its symmetry axis is given by I = (1/2) MR², then the maximum acceleration the cylinder can have without sliding is:

A 5.0-kg ball rolls without sliding from rest down an inclined plane. A 4.0-kg block, mounted on roller bearings totaling 100 g, rolls from rest down the same plane. At the bottom, the block has:

A yo-yo, arranged as shown, rests on a frictionless surface. When a force is applied to the string as shown, the yo-yo:

When the speed of a rear-drive car is increasing on a horizontal road the direction of the frictional force on the tires is:

A forward force acting on the axle accelerates a rolling wheel on a horizontal surface. If the wheel does not slide the frictional force of the surface on the wheel is:

A single force acts on a particle situated on the positive x axis. The torque about the origin is in the negative z direction. The force might be:

Two uniform cylinders have different masses and different rotational inertias. They simultaneously start from rest at the top of an inclined plane and roll without sliding down the plane. The cylinder that gets to the bottom first is:

When we apply the energy conversation principle to a cylinder rolling down an incline without sliding, we exclude the work done by friction because:

A thin-walled hollow tube rolls without sliding along the floor. The ratio of its translational kinetic energy to its rotational kinetic energy (about an axis through its center of mass) is:

A sphere and a cylinder of equal mass and radius are simultaneously released from rest on the same inclined plane sliding down the incline. Then:

Two wheels roll side-by-side without sliding, at the same speed. The radius of wheel 2 is twice the radius of wheel 1. The angular velocity of wheel 2 is:

A hoop, a uniform disk, and a uniform sphere, all with the same mass and outer radius, start with the same speed and roll without sliding up identical inclines. Rank the objects according to how high they go, least to greatest.

A hoop rolls with constant velocity and without sliding along level ground. Its ratation kinetic energy is: