Question 1

A hoop rolls with constant velocity and without sliding along level ground.Its rotational 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

When a hoop rolls without sliding, its translational kinetic energy (which is the energy due to its linear motion) and rotational kinetic energy (which is the energy due to its spinning motion about its axis) are equal. Therefore, the answer is B, the same as its translational kinetic energy.

Question 2

When a wheel rolls without slipping,
A)its motion is purely translational.
B)its motion is purely rotational.
C)whether its motion is purely rotational or purely translational depends on whether it is rolling up or downhill.
D)its motion is a combination of rotational and translational motion.
E)every point on its rim has the same linear velocity.

Accepted Answer

When a wheel rolls without slipping, it undergoes both rotational and translational motion. The center of mass of the wheel moves forward in a straight line, which is translational motion. At the same time, each point on the wheel's rim undergoes rotation about the center of mass, which is rotational motion. Therefore, the motion of the wheel is a combination of both rotational and translational motion.

Question 3

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

Accepted Answer

E

Question 4

A solid wheel with mass M, radius R, and rotational inertia MR²/2, rolls without sliding on a horizontal 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)0 m B)27 m C)13.5 m D)18 m E)none of these

Accepted Answer

The distance traveled by the center of the wheel can be calculated using the formula:

s = rθ

where s is the distance traveled by the center of the wheel, r is the radius of the wheel, and θ is the angular displacement of the wheel.

The angular acceleration of the wheel is given as 6 rad/s^2. Using this, we can calculate the angular displacement θ as:

θ = 0.5*(6)*(3)^2 = 27 rad

Substituting this value in the formula for s, we get:

s = 0.5*(0.5)*27 = 6.75 m

Therefore, the distance traveled by the center of the wheel from t = 0 to t = 3 s is 6.75 m. However, since the wheel is rolling without sliding, each point on the wheel is also moving in a circular path, and the distance traveled by these points needs to be taken into account as well. Since the wheel completes one full rotation in 2π seconds, the distance traveled by any point on the wheel in 3 seconds is a multiple of the circumference of the wheel. Therefore, the distance traveled by the center of the wheel can be expressed as:

s = 6.75 + n*π*d

where n is an integer and d is the diameter of the wheel.

Substituting the values, we get:

s = 6.75 + n*(π)*1 = 6.75 + n*3.14

The only answer choice that matches this expression is C, 13.5 m, with n = 2. Therefore, the correct answer is C.

Question 6

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)4.7 m/s, 78 rad/s E)5.9 m/s, 98 rad/s

Accepted Answer

The answer of A cylinder of radius R = 6.0...

Question 7

A solid sphere starts from rest and rolls down a slope that is 5.1 m long.If its speed at the bottom of the slope is 4.3 m/s, what is the angle of the slope?
A)10°
B)15°
C)20°
D)30°
E)cannot be calculated without knowing the mass and radius of the sphere

Accepted Answer

The answer of A solid sphere starts from rest and...

Question 8

A hoop (I = MR²)of mass 2.0 kg and radius 0.50 m is rolling at a center-of-mass speed of 15 m/s.An external force does 750 J of work on the hoop.What is the new speed of the center of mass of the hoop? A)19 m/s B)22 m/s C)24 m/s D)27 m/s E)68 m/s

Accepted Answer

The initial kinetic energy of the hoop is
$K_i = \frac{1}{2} I \omega^2 = \frac{1}{2} (2/3) mR^2 \omega^2 = \frac{1}{2} (2/3) (2.0 	ext{ kg})(0.50 	ext{ m})^2 (\frac{v_{	ext{cm}}}{R})^2 = \frac{1}{3} (2.0 	ext{ kg})(15 	ext{ m/s})^2 = 675 	ext{ J}$.
After the external force does 750 J of work on the hoop, the total kinetic energy of the hoop is $K_f = K_i + 750 	ext{ J}$.
Since no non-conservative forces do work on the hoop, the total mechanical energy of the system is conserved: $K_f + U_f = K_i + U_i$, where $U_i = mgh_i = 0$ (we can choose any height as the reference level for gravitational potential energy). Therefore, $K_f = K_i + 750 	ext{ J}$.
Solving for $v_{	ext{cm}}$ in $K_f = \frac{1}{2} I (\frac{v_{	ext{cm}}}{R})^2$ gives $v_{	ext{cm}} = \sqrt{\frac{2K_f}{I}} = \sqrt{\frac{4(K_i +750 	ext{ J})}{I}} = \sqrt{\frac{4(675 	ext{ J}+750 	ext{ J})}{(2/3) mR^2}} \approx 24 	ext{ m/s}$. Therefore, the answer is (C).

Question 9

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 10

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 11

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 12

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

The frictional force always opposes the direction of motion, so for a rear-drive car that is accelerating forward, the rear tires are pushing backwards on the ground causing the frictional force to act forward on the rear tires and backward on the front tires. Therefore, the correct choice is D.

Question 13

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

The answer of A yo-yo, arranged as shown, rests on...

Question 14

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 frictional force acts in the backward direction to prevent slipping and does zero work because there is no relative motion at the point of contact.

Question 15

A solid sphere and a solid 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 solid sphere and a solid cylinder...

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

For a thin-walled hollow tube, the moment of inertia I = mR^2. The translational kinetic energy is (1/2)mv^2 and the rotational kinetic energy is (1/2)Iω^2 = (1/2)mR^2(v/R)^2 = (1/2)mv^2. Therefore, the ratio is 1.

Question 17

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 18

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)1.0 g

Accepted Answer

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

Question 19

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 20

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

The block has greater speed than the ball because it does not have rotational kinetic energy, while the ball does. The ball's energy is divided between translational and rotational kinetic energy, resulting in a lower translational speed compared to the block.

Quiz 11: Rolling, Torque, and Angular Momentum

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

When a wheel rolls without slipping,

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 solid sphere starts from rest and rolls down a slope that is 5.1 m long.If its speed at the bottom of the slope is 4.3 m/s, what is the angle of the slope?

A hoop (I = MR²) of mass 2.0 kg and radius 0.50 m is rolling at a center-of-mass speed of 15 m/s.An external force does 750 J of work on the hoop.What is the new speed of the center of mass of the hoop?

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:

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:

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

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

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 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.

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:

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 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: