Question 1

A 60.0-kg skater begins a spin with an angular speed of 6.0 rad/s. By changing the position of her arms, the skater decreases her moment of inertia to one-half its initial value. What is the skater's final angular speed?

Accepted Answer

The answer of A 60.0-kg skater begins a spin with...

Question 2

A 2.0-kg hoop rolls without slipping on a horizontal surface so that its center proceeds to the right with a constant linear speed of 6.0 m/s. 
-What is the total kinetic energy of the hoop?

Accepted Answer

Since the hoop is moving with a constant linear speed, its kinetic energy is only due to its rotational KE. The equation for rotational KE is:
KE(rot) = (1/2)*I*w^2
where I is the moment of inertia and w is the angular velocity.
For a hoop, I = MR^2, where M is the mass and R is the radius.
Since the hoop is rolling without slipping, w = v/R, where v is the linear speed.
Substituting the values given in the problem:
KE(rot) = (1/2)*MR^2*(v/R)^2 = (1/2)*M*v^2
= (1/2)*2.0 kg*(6.0 m/s)^2 = 72 J
Therefore, the correct answer is (C)

Question 3

What happens when a spinning ice skater draws in her outstretched arms?

Accepted Answer

When the skater draws in her arms, her moment of inertia decreases, and to conserve angular momentum, her rotational speed increases.

Question 4

Two skaters, each of mass 40 kg, approach each other along parallel paths that are separated by a distance of 2 m. Both skaters have a speed of 10 m/s. The first skater carries a 2-m pole that may be considered massless. As he passes the pole, the second skater catches hold of the end. The two skaters then go around in a circle about the center of the pole. 
-What is their combined angular momentum about the center of the pole?

Accepted Answer

The answer of Two skaters, each of mass 40 kg,...

Question 5

A 2.0-kg hoop rolls without slipping on a horizontal surface so that its center proceeds to the right with a constant linear speed of 6.0 m/s. 
-Which one of the following statements is true concerning the angular momentum of this hoop?

Accepted Answer

The angular momentum of a hoop rolling without slipping is directed along the axis of rotation, which is perpendicular to the plane of the hoop and points into the paper.

Question 6

A solid sphere of radius R rotates about a diameter with an angular speed $\omega$. The sphere then collapses under the action of internal forces to a final radius R/2. What is the final angular speed of the sphere?

Accepted Answer

The moment of inertia of a solid sphere about a diameter is $I = \frac{2}{5}mR^2$. When the sphere collapses to a radius of R/2, its new moment of inertia becomes $I' = \frac{2}{5}m(R/2)^2 = \frac{1}{4} \times \frac{2}{5}mR^2$. By conservation of angular momentum, $I\omega = I'\omega'$. Solving for $\omega'$, we get $\omega' = 4\omega$.

Question 7

A spinning star begins to collapse under its own gravitational pull. Which one of the following occurs as the star becomes smaller?

Accepted Answer

Angular momentum is a conserved quantity, meaning it remains constant unless acted upon by an external torque. As the spinning star collapses, its radius decreases, which causes its angular velocity to increase in order to conserve its angular momentum. Therefore, (A), (C), and (D) can be eliminated. Since angular momentum is conserved, (E) can also be eliminated. Therefore, the correct answer is (B) as the star's angular momentum remains constant during its collapse.

Question 8

Two skaters, each of mass 40 kg, approach each other along parallel paths that are separated by a distance of 2 m. Both skaters have a speed of 10 m/s. The first skater carries a 2-m pole that may be considered massless. As he passes the pole, the second skater catches hold of the end. The two skaters then go around in a circle about the center of the pole. 
-What is the angular speed of the skaters after they have linked together?

Accepted Answer

The answer of Two skaters, each of mass 40 kg,...

Question 9

Planets A and B are uniform solid spheres that rotate at a constant speed about axes through their centers. Although B has twice the mass and three times the radius of A, each planet has the same rotational kinetic energy. What is the ratio $\omega$_B/$\omega$_A of their angular speeds?

Accepted Answer

The answer of Planets A and B are uniform solid...

Question 10

A 1500-kg satellite orbits a planet in a circular orbit of radius 6.2 × 10⁶ m. What is the angular momentum of the satellite in its orbit around the planet if the satellite completes one orbit every 1.5 × 10⁴ s?

Accepted Answer

The angular momentum L of a satellite in a circular orbit is given by L = mvr, where m is the mass, v is the orbital speed, and r is the radius. The orbital speed v can be calculated using v = 2πr/T, where T is the period. Substituting the given values, L = 1500 kg * (2π * 6.2 × 10^6 m / 1.5 × 10^4 s) * 6.2 × 10^6 m = 2.4 × 10^13 kg . m^2/s.

Question 11

A uniform disk of radius 1.2 m and mass 0.60 kg is rotating at 25 rad/s around an axis that passes through its center and is perpendicular to the disk. A rod makes contact with the rotating disk with a force of 4.5 N at a point 0.75 m from the axis of rotation as shown. The disk is brought to a stop in 5.0 s. What is the coefficient of kinetic friction for the two materials in contact?

Accepted Answer

First, we need to use the work-energy principle. The work done by the rod is equal to the change in kinetic energy of the disk:
$W = \Delta K
$
$W = \frac{1}{2} I \omega_f^2 - \frac{1}{2} I \omega_i^2
$
where $I$ is the moment of inertia of the disk, $\omega_i$ is the initial angular velocity, and $\omega_f$ is the final angular velocity (zero, since the disk comes to a stop). Substituting in the given values, we get:
$4.5 	ext{ N} \cdot 0.75 	ext{ m} = \frac{1}{2} (0.5 m r^2) (0 - (25 	ext{ rad/s})^2)
$
Simplifying and solving for $r$, we get:
$r = 1.2 	ext{ m}
$.
Next, we can use the equation for the frictional torque on a rotating disk:
$	au_f = I \alpha
$
where $	au_f$ is the frictional torque, $I$ is the moment of inertia of the disk, and $\alpha$ is the angular acceleration. Substituting in the given values, we get:
$	au_f = (0.5 m r^2) \left(-\frac{\omega_i}{t}ight)
$
where $t$ is the time it takes for the disk to come to a stop (5.0 s). Solving for $	au_f$, we get:
$	au_f = 0.75 	ext{ N} \cdot 	ext{m}
$.
Finally, we can use the equation for the coefficient of kinetic friction:
$\mu_k = \frac{	au_f}{r F}
$
where $F$ is the force applied by the rod. Substituting in the given values, we get:
$\mu_k = \frac{0.75 	ext{ N} \cdot 	ext{m}}{(1.2 	ext{ m})(4.5 	ext{ N})}
$
Simplifying, we get:
$\mu_k = 0.64
$
Therefore, the coefficient of kinetic friction for the two materials in contact is 0.64, which corresponds to choice C.

Question 12

A spinning skater draws in her outstretched arms thereby reducing her moment of inertia by a factor of 2. Determine the ratio of her final kinetic energy to her initial kinetic energy.

Accepted Answer

According to the law of conservation of angular momentum, the product of moment of inertia and angular velocity must remain constant. As moment of inertia reduces by a factor of 2, angular velocity doubles to maintain angular momentum. Kinetic energy of a rotating object is given by (1/2)*I*w^2. As w doubles, kinetic energy increases by a factor of 2^2 = 4. Therefore, the ratio of final to initial kinetic energy is 4/1 = 4, which corresponds to option (C).

Question 13

A 4.0-kg ball moves in a straight line at 8.0 m/s as shown in the figure. At the instant shown, what is its angular momentum about the point P?

Accepted Answer

The answer of A 4.0-kg ball moves in a straight...

Question 14

A steady horizontal force  of magnitude 21 N is applied at the axle of a solid disk as shown. The disk has mass 2.0 kg and diameter 0.10 m. What is the linear speed of the center of the disk after it has moved 12 m?

Accepted Answer

The answer of A steady horizontal force  of magnitude...

Question 15

Two equal spheres, labeled A and B in the figure, are attached to a massless rod with a frictionless pivot at the point P. The system is made to rotate clockwise with angular speed $\omega$on a horizontal, frictionless tabletop. Sphere A collides with and sticks to another equal sphere that is at rest on the tabletop. Note: the masses of all three spheres are equal. What is the angular speed of the system immediately after the collision?

Accepted Answer

The answer of Two equal spheres, labeled A and B...

Question 16

A ball of mass M moves in a circular path on a horizontal, frictionless surface. It is attached to a light string that passes through a hole in the center of the table. If the string is pulled down, thereby reducing the radius of the path of the ball, the speed of the ball is observed to increase. Complete the following sentence: This occurs because

Accepted Answer

As the radius of the circular motion decreases, the linear velocity of the ball must increase in order to maintain constant angular velocity. This means that the angular momentum of the ball (L = Iw) must remain constant, as the moment of inertia of the ball does not change. Therefore, the correct choice is C, the angular momentum of the ball is conserved.

Question 17

A compact disc rotates about its center at constant angular speed. Which one of the following quantities is constant and non-zero for a dust particle near the edge of the disc?

Accepted Answer

Angular momentum is conserved for a dust particle near the edge of the disc because there is no external torque acting on the particle.

Quiz 9: Rotational Dynamics

A 60.0-kg skater begins a spin with an angular speed of 6.0 rad/s. By changing the position of her arms, the skater decreases her moment of inertia to one-half its initial value. What is the skater's final angular speed?

A 2.0-kg hoop rolls without slipping on a horizontal surface so that its center proceeds to the right with a constant linear speed of 6.0 m/s. -What is the total kinetic energy of the hoop?

What happens when a spinning ice skater draws in her outstretched arms?

A 2.0-kg hoop rolls without slipping on a horizontal surface so that its center proceeds to the right with a constant linear speed of 6.0 m/s. -Which one of the following statements is true concerning the angular momentum of this hoop?

A solid sphere of radius R rotates about a diameter with an angular speed $\omega$ . The sphere then collapses under the action of internal forces to a final radius R/2. What is the final angular speed of the sphere?

A spinning star begins to collapse under its own gravitational pull. Which one of the following occurs as the star becomes smaller?

Planets A and B are uniform solid spheres that rotate at a constant speed about axes through their centers. Although B has twice the mass and three times the radius of A, each planet has the same rotational kinetic energy. What is the ratio $\omega$ _B/ $\omega$ _A of their angular speeds?

A 1500-kg satellite orbits a planet in a circular orbit of radius 6.2 × 10⁶ m. What is the angular momentum of the satellite in its orbit around the planet if the satellite completes one orbit every 1.5 × 10⁴ s?

A spinning skater draws in her outstretched arms thereby reducing her moment of inertia by a factor of 2. Determine the ratio of her final kinetic energy to her initial kinetic energy.

A 4.0-kg ball moves in a straight line at 8.0 m/s as shown in the figure. At the instant shown, what is its angular momentum about the point P?

A steady horizontal force of magnitude 21 N is applied at the axle of a solid disk as shown. The disk has mass 2.0 kg and diameter 0.10 m. What is the linear speed of the center of the disk after it has moved 12 m?

A compact disc rotates about its center at constant angular speed. Which one of the following quantities is constant and non-zero for a dust particle near the edge of the disc?

Quiz 9: Rotational Dynamics

A 60.0-kg skater begins a spin with an angular speed of 6.0 rad/s. By changing the position of her arms, the skater decreases her moment of inertia to one-half its initial value. What is the skater's final angular speed?

A 2.0-kg hoop rolls without slipping on a horizontal surface so that its center proceeds to the right with a constant linear speed of 6.0 m/s. -What is the total kinetic energy of the hoop?

What happens when a spinning ice skater draws in her outstretched arms?

A 2.0-kg hoop rolls without slipping on a horizontal surface so that its center proceeds to the right with a constant linear speed of 6.0 m/s. -Which one of the following statements is true concerning the angular momentum of this hoop?

A solid sphere of radius R rotates about a diameter with an angular speed ω\omegaω . The sphere then collapses under the action of internal forces to a final radius R/2. What is the final angular speed of the sphere?

A spinning star begins to collapse under its own gravitational pull. Which one of the following occurs as the star becomes smaller?

Planets A and B are uniform solid spheres that rotate at a constant speed about axes through their centers. Although B has twice the mass and three times the radius of A, each planet has the same rotational kinetic energy. What is the ratio ω\omegaω B/ ω\omegaω A of their angular speeds?

A 1500-kg satellite orbits a planet in a circular orbit of radius 6.2 × 106 m. What is the angular momentum of the satellite in its orbit around the planet if the satellite completes one orbit every 1.5 × 104 s?

A spinning skater draws in her outstretched arms thereby reducing her moment of inertia by a factor of 2. Determine the ratio of her final kinetic energy to her initial kinetic energy.

A 4.0-kg ball moves in a straight line at 8.0 m/s as shown in the figure. At the instant shown, what is its angular momentum about the point P?

A steady horizontal force of magnitude 21 N is applied at the axle of a solid disk as shown. The disk has mass 2.0 kg and diameter 0.10 m. What is the linear speed of the center of the disk after it has moved 12 m?

A compact disc rotates about its center at constant angular speed. Which one of the following quantities is constant and non-zero for a dust particle near the edge of the disc?

A solid sphere of radius R rotates about a diameter with an angular speed $\omega$ . The sphere then collapses under the action of internal forces to a final radius R/2. What is the final angular speed of the sphere?

Planets A and B are uniform solid spheres that rotate at a constant speed about axes through their centers. Although B has twice the mass and three times the radius of A, each planet has the same rotational kinetic energy. What is the ratio $\omega$ _B/ $\omega$ _A of their angular speeds?

A 1500-kg satellite orbits a planet in a circular orbit of radius 6.2 × 10⁶ m. What is the angular momentum of the satellite in its orbit around the planet if the satellite completes one orbit every 1.5 × 10⁴ s?