Question 1

A particle moves uniformly around the circumference of a circle whose radius is 8.0 cm with a period of /20 s. The angular velocity of the particle is A) 2.5 rad/s. B) 3.2  10² rad/s. C) 40 rad/s. D) 7.9 rad/s. E) 0.96 rad/s.

Accepted Answer

The angular velocity (ω) is given by the formula ω = 2π/T, where T is the period. Given T = 20 s, ω = 2π/20 = π/10 ≈ 0.314 rad/s. However, the period seems to be misrepresented in the question, as the correct calculation should yield 40 rad/s, which matches choice C.

Question 2

Two objects, m₁ and m₂, both of mass m, are place on a horizontal platform that is rotating at a constant angular velocity. m₁ = m is located at a distance R from the axis of rotation and the second object of mass m₂ = 2m is located at a distance 2R. The angular velocity of mass m₁____ to the angular velocity of m₂. A) is less than B) is equal to C) is greater than D) depends how fast it is rotating E) Unable to tell.

Accepted Answer

The angular velocity is constant for all objects on the rotating platform. It does not depend on the distance from the axis of rotation or the mass of the object. Therefore, the angular velocity of m₁ is equal to the angular velocity of m₂.

Question 3

Four 50-g point masses are at the corners of a square with 20-cm sides. What is the moment of inertia of this system about an axis perpendicular to the plane of the square and passing through its center? A) 1.0  10^-³ kg · m² B) 4.0  10^-³ kg · m² C) 2.0  10^-³ kg · m² D) 8.0  10^-³ kg · m² E) 2.8  10^-³ kg · m²

Accepted Answer

B

Question 4

Two objects, m₁ and m₂, both of mass m, are placed on a horizontal platform that is rotating at a constant angular velocity. m₁ is located at a distance R from the axis of rotation and m₂ is located at a R. The centripetal acceleration of mass m₁ ____ to the centripetal acceleration of m₂. A) is less than B) is equal to C) is greater than D) depends how fast it is rotating E) Unable to tell.

Accepted Answer

The answer of Two objects, m₁ and m₂, both of...

Question 5

A hoop of mass 50 kg rolls without slipping. If the center of mass of the hoop has a translational speed of 4.0 m/s, the total kinetic energy of the hoop is
A) 0.20 kJ.
B) 0.40 kJ.
C) 1.1 kJ.
D) 3.9 kJ.
E) None of these is correct.

Accepted Answer

The total kinetic energy of a rolling object (without slipping) is the sum of its translational kinetic energy and rotational kinetic energy. The translational kinetic energy ($K_{trans}$) is given by $\frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity of the center of mass. The rotational kinetic energy ($K_{rot}$) about the center of mass for a hoop is given by $\frac{1}{2}I\omega^2$, where $I$ is the moment of inertia of the hoop about its center ($I = mr^2$ for a hoop) and $\omega$ is the angular velocity. Since the hoop rolls without slipping, $v = r\omega$, where $r$ is the radius of the hoop.Given: $m = 50\,kg$, $v = 4.0\,m/s$1. Calculate $K_{trans} = \frac{1}{2}mv^2 = \frac{1}{2} \times 50 \times (4.0)^2 = 400\,J$2. To find $K_{rot}$, we use $v = r\omega$ to express $\omega$ as $\omega = \frac{v}{r}$. Then, $K_{rot} = \frac{1}{2}I\omega^2 = \frac{1}{2} \times (mr^2) \times \left(\frac{v}{r}\right)^2 = \frac{1}{2} \times 50 \times 4^2 = 400\,J$Thus, the total kinetic energy $K_{total} = K_{trans} + K_{rot} = 400\,J + 400\,J = 800\,J$, which is not listed among the options provided.

Question 6

You have a friend who lives in the southern part of the United States, and you live in the northern part. As Earth rotates, your linear velocity is ___________ hers, and your angular velocity is ____________ hers.
A) greater than; equal to
B) equal to; greater than
C) greater than; less than
D) less than; greater than
E) less than; equal to

Accepted Answer

The answer of You have a friend who lives in...

Question 7

The moment of inertia of a slim rod about a transverse crosswise) axis through one end is mL²/3, where m is the mass of the rod and L is its length. The moment of inertia of a 0.24-kg meter stick about a transverse axis through its center is A) 0.14 kg · m². B) 20 kg · m². C) 0.020 kg · m². D) 80 kg · m². E) 4.5 kg · m².

Accepted Answer

The given moment of inertia formula for a slim rod is:
$I = \frac{1}{3} mL^2$
For a meter stick of mass 0.24 kg and length 1 m, the formula becomes:
$I = \frac{1}{3} (0.24	ext{ kg})(1	ext{ m})^2 = 0.080	ext{ kg}\cdot	ext{m}^2$
But this is the moment of inertia about the center axis, not the end. We can use the parallel axis theorem to find the moment of inertia about an axis through one end:
$I_{end} = I_{center} + Mh^2$
Where $M$ is the mass of the meter stick, $h$ is the distance from the end to the center of mass (which is half the length of the meter stick), and we can substitute $I_{center}$ as found above:
$I_{end} = 0.080	ext{ kg}\cdot	ext{m}^2 + (0.24	ext{ kg})(0.5	ext{ m})^2 = 0.020	ext{ kg}\cdot	ext{m}^2$
Therefore, the answer is choice C.

Question 8

A hoop of radius 3.05 m has a mass of 145 kg. Its moment of inertia is mR². The hoop rolls without slipping along a horizontal plane. If the center of mass of the hoop has a speed of 0.305 m/s, the work required to bring the hoop to rest is A) 6.78 J. B) 13.5 J. C) 682 J. D) 217 J. E) 4.34 kJ.

Accepted Answer

The total kinetic energy of the hoop is the sum of translational and rotational kinetic energy. The work required to bring the hoop to rest is equal to this total kinetic energy. Calculating it gives 13.5 J.

Question 9

The moment of inertia of a set of dumbbells, considered as two mass points m separated by a distance 2L about the axis A, is A) mL². B) mL². C) 2mL². D) mL². E) 4mL².

Accepted Answer

The moment of inertia of a set of dumbbells, considered as two mass points m separated by a distance 2L about the axis A, is given by the formula: I = 2mL^2 So, substituting for L, we get: I = 2m(^-)^2 Expanding the square, we get: I = 2m(^{^2 - 2 +}^2) Simplifying, we get: I = 2m(².⁺) Therefore, the correct answer is C.

Question 10

A homogeneous solid cylinder of mass m, length L, and radius R rotates about an axis through point P, which is parallel to the cylinder axis. If the moment of inertia about the cylinder axis is mR², the moment of inertia about the axis through P is A) 0.4mR². B) mR². C) 2/3 mR². D) mR². E) 1.5mR².

Accepted Answer

The answer of  A homogeneous solid cylinder of mass...

Question 11

A wheel on a car is rolling without slipping along level ground. The speed of the car is 36 m/s. The wheel has an outer diameter of 50 cm. The speed of the top of the wheel is
A) 36 m/s.
B) 3.6 m/s.
C) 72 m/s.
D) 18 m/s.
E) 98 m/s.

Accepted Answer

The answer of A wheel on a car is rolling...

Question 12

The London Eye, a giant Ferris wheel, has a diameter of 135 m. It revolves at a constant rate and takes 30 minutes to complete one revolution. What is the linear velocity of a rider in a capsule that is located at the perimeter of the wheel?
A) 0.24 m/s
B) 0.47 m/s
C) 3.5 m/s
D) 7.1 m/s
E) None of the above.

Accepted Answer

The answer of The London Eye, a giant Ferris wheel,...

Question 13

The moment of inertia of a slim rod of mass m and length L about a transverse axis through one end is mL²/3. The moment of inertia of such a rod about a transverse axis through the rod at a distance L/3 from one end is A) mL²/36. B) 7mL²/36. C) mL²/9. D) 2mL²/9. E) 4mL²/9.

Accepted Answer

The moment of inertia of a slim rod of mass $m$ and length $L$ about a transverse axis through one end is given by:
$$I = \frac{1}{3}mL^2$$
Using this formula, we find that the moment of inertia of the rod about a transverse axis through a distance $L/3$ from one end is:
$$I' = \frac{1}{3}m\left(\frac{L}{3}\right)^2 + m\left(\frac{2L}{3}\right)^2 = \frac{4}{9}mL^2$$
Substituting the given formula for the moment of inertia about a transverse axis through one end, we have:
$$\frac{1}{3}mL^2 = \frac{mL}{2}\left(\frac{L}{3}\right)^2 + I''$$
where $I''$ is the moment of inertia about a transverse axis through the center of mass of the rod. Solving for $I''$, we get:
$$I'' = \frac{1}{3}mL^2 - \frac{mL}{18} = \frac{5}{18}mL^2$$
Using the parallel axis theorem, the moment of inertia about a transverse axis through a distance $L/3$ from one end is:
$$I' = I'' + md^2 = \frac{5}{18}mL^2 + \frac{mL^2}{9} = \frac{7}{18}mL^2$$
Simplifying the expression, we get:
$$I' = \frac{1}{9}mLS_1U_1P_{12}S_1S_1P_0$$
Therefore, the answer is (C).

Question 14

A wheel of radius R is rolling without slipping. The velocity of the point on the rim that is in contact with the surface, relative to the surface, is
A) equal to R in the direction of motion of the center of mass.
B) equal to R opposite the direction of motion of the center of mass.
C) zero.
D) equal to the velocity of the center of mass and in the same direction.
E) equal to the velocity of the center of mass but in the opposite direction.

Accepted Answer

When a wheel rolls without slipping, the point on the rim in contact with the surface has zero velocity relative to the surface.

Question 15

A wheel is rotating at 30 rev/min. The angular velocity of the wheel is A) 2π² rad/s. B) 2π rad/s. C) 2 rad/s. D) π/2 rad/s. E) π rad/s.

Accepted Answer

Angular velocity (ω) is given by:

ω = 2πn

where n is the number of revolutions per unit time.

Substituting n = 30 rev/min,

ω = 2π (30 rev/min)

= 60π rad/min

= π rad/s

Therefore, the angular velocity of the wheel is π rad/s. The correct answer is E.

Question 16

To increase the moment of inertia of a body about an axis, you must
A) increase the angular acceleration.
B) increase the angular velocity.
C) decrease the angular velocity.
D) make the body occupy less space.
E) place part of the body farther from the axis.

Accepted Answer

The moment of inertia of a body is directly proportional to the distance from the axis of rotation. Therefore, placing part of the body farther from the axis will increase its moment of inertia. Choice A is incorrect because increasing angular acceleration will only change the rate at which the body's angular velocity changes, but not its moment of inertia. Choice B is incorrect because increasing the body's angular velocity will have no effect on its moment of inertia. Choice C is incorrect because decreasing the body's angular velocity will have no effect on its moment of inertia. Choice D is incorrect because the amount of space that a body occupies has no direct relationship with its moment of inertia.

Question 17

The disc brake of a high performance car is often made of carbon fiber instead of iron, thereby reducing the mass. If the mass of a carbon disc is that of iron for the same shape and size, by what factor is the moment of inertia changed?
A) one quarter
B) half
C) unchanged
D) doubled
E) quadrupled

Accepted Answer

The moment of inertia is proportional to the mass. If the mass is reduced by a factor of 4, the moment of inertia will also be reduced by a factor of 4.

Question 18

A particle is moving uniformly in a circle of radius 50 cm. Its angular velocity is 96 rad/s. The linear speed of the particle is
A) 1.0 m/s.
B) 96 cm/s.
C) 48 m/s.
D) zero.
E) 15 m/s.

Accepted Answer

The answer of A particle is moving uniformly in a...

Question 19

A body of mass m is whirled at a constant angular velocity on the end of a string of length R. To double the kinetic energy of the body as it whirls while maintaining the angular velocity, the length of the string must be changed to
A) 2R.
B)  .
C) R/2.
D) 4R.
E)  .

Accepted Answer

The answer of A body of mass m is whirled...

Question 20

Michelle Kwan, the American figure skater, is spinning on ice with her arms extended. She then draws her arms in and crosses them in front of her chest. How has her moment of inertia changed?
A) Her moment of inertia has decreased.
B) Her moment of inertia has increased.
C) Her moment of inertia has not changed.
D) Her moment of inertia decreases as she draws her arms in, but then returns to the initial value once she has crossed them.
E) Her moment of inertia increases as she draws her arms in, but then returns to the initial value once she has crossed them.

Accepted Answer

When Michelle Kwan draws her arms in, her mass is closer to the axis of rotation, which decreases her moment of inertia.

Quiz 8: Rotational Motion

A particle moves uniformly around the circumference of a circle whose radius is 8.0 cm with a period of /20 s. The angular velocity of the particle is

Four 50-g point masses are at the corners of a square with 20-cm sides. What is the moment of inertia of this system about an axis perpendicular to the plane of the square and passing through its center?

A hoop of mass 50 kg rolls without slipping. If the center of mass of the hoop has a translational speed of 4.0 m/s, the total kinetic energy of the hoop is

You have a friend who lives in the southern part of the United States, and you live in the northern part. As Earth rotates, your linear velocity is _ hers, and your angular velocity is __ hers.

The moment of inertia of a slim rod about a transverse crosswise) axis through one end is mL²/3, where m is the mass of the rod and L is its length. The moment of inertia of a 0.24-kg meter stick about a transverse axis through its center is

A hoop of radius 3.05 m has a mass of 145 kg. Its moment of inertia is mR². The hoop rolls without slipping along a horizontal plane. If the center of mass of the hoop has a speed of 0.305 m/s, the work required to bring the hoop to rest is

The moment of inertia of a set of dumbbells, considered as two mass points m separated by a distance 2L about the axis A, is

A homogeneous solid cylinder of mass m, length L, and radius R rotates about an axis through point P, which is parallel to the cylinder axis. If the moment of inertia about the cylinder axis is mR², the moment of inertia about the axis through P is

A wheel on a car is rolling without slipping along level ground. The speed of the car is 36 m/s. The wheel has an outer diameter of 50 cm. The speed of the top of the wheel is

The London Eye, a giant Ferris wheel, has a diameter of 135 m. It revolves at a constant rate and takes 30 minutes to complete one revolution. What is the linear velocity of a rider in a capsule that is located at the perimeter of the wheel?

The moment of inertia of a slim rod of mass m and length L about a transverse axis through one end is mL²/3. The moment of inertia of such a rod about a transverse axis through the rod at a distance L/3 from one end is

A wheel of radius R is rolling without slipping. The velocity of the point on the rim that is in contact with the surface, relative to the surface, is

A wheel is rotating at 30 rev/min. The angular velocity of the wheel is

To increase the moment of inertia of a body about an axis, you must

The disc brake of a high performance car is often made of carbon fiber instead of iron, thereby reducing the mass. If the mass of a carbon disc is that of iron for the same shape and size, by what factor is the moment of inertia changed?

A particle is moving uniformly in a circle of radius 50 cm. Its angular velocity is 96 rad/s. The linear speed of the particle is

A body of mass m is whirled at a constant angular velocity on the end of a string of length R. To double the kinetic energy of the body as it whirls while maintaining the angular velocity, the length of the string must be changed to

Michelle Kwan, the American figure skater, is spinning on ice with her arms extended. She then draws her arms in and crosses them in front of her chest. How has her moment of inertia changed?