System of Particles and Rotational Motion Chapter-Wise Test 3

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A solid cylinder of mass \( 6 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) has an angular momentum of \( 10.8 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = \frac{1}{2} M R^2 = \frac{1}{2} \times 6 \times (0.3)^2 = 0.27 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{10.8}{0.27} = 40 \, \text{rad/s} \).

35 rad/s
40 rad/s
45 rad/s
50 rad/s
2

What is the physical significance of the moment of inertia in rotational motion?

Moment of inertia measures a body’s resistance to angular acceleration, analogous to mass in linear motion, and depends on mass distribution.

It measures the speed of rotation
It measures resistance to angular acceleration
It measures the force applied
It measures the angular displacement
2

For a rigid body to be in rotational equilibrium, what condition must be satisfied?

For rotational equilibrium, the net torque about any point must be zero (\( \sum \mathbf{\tau} = 0 \)), meaning no angular acceleration occurs.

The net force must be zero
The net torque must be zero
The angular velocity must be zero
The moment of inertia must be zero
2

Which of the following correctly describes the rotational kinetic energy of a rigid body?

Rotational kinetic energy is given by \( K = \frac{1}{2} I \omega^2 \), depending on moment of inertia and angular velocity, not linear velocity directly.

\( K = \frac{1}{2} m v^2 \)
\( K = \frac{1}{2} I \omega^2 \)
\( K = I \omega \)
\( K = m \omega^2 \)
2

A solid cylinder of mass \( 3 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) rotates at \( 7 \, \text{rad/s} \). What is its angular momentum about its axis?

\( I = \frac{1}{2} M R^2 = \frac{1}{2} \times 3 \times (0.5)^2 = 0.375 \, \text{kg m}^2 \).

\( L = I \omega = 0.375 \times 7 = 2.625 \, \text{kg m}^2/\text{s} \).

2.5 kg m²/s
2.625 kg m²/s
2.8 kg m²/s
3.0 kg m²/s
2

A torque of \( 18 \, \text{Nm} \) acts on a wheel with moment of inertia \( 9 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 6 \, \text{s} \)?

\( \alpha = \frac{\tau}{I} = \frac{18}{9} = 2 \, \text{rad/s}^2 \).

\( \omega = \omega_0 + \alpha t = 0 + 2 \times 6 = 12 \, \text{rad/s} \).

10 rad/s
11 rad/s
12 rad/s
13 rad/s
3

A particle rotates about a fixed axis at a radius of \( 1.2 \, \text{m} \) with an angular velocity of \( 4 \, \text{rad/s} \). What is its linear speed?

\( v = \omega r \).

\( \omega = 4 \, \text{rad/s} \), \( r = 1.2 \, \text{m} \).

\( v = 4 \times 1.2 = 4.8 \, \text{m/s} \).

4.5 m/s
4.8 m/s
5.0 m/s
5.2 m/s
2

A hollow cylinder of mass \( 6 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) has an angular momentum of \( 10.8 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = M R^2 = 6 \times (0.3)^2 = 0.54 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{10.8}{0.54} = 20 \, \text{rad/s} \).

18 rad/s
19 rad/s
20 rad/s
21 rad/s
3

What is the physical meaning of angular momentum in rotational motion?

Angular momentum (\( \mathbf{L} = I \mathbf{\omega} \)) quantifies the rotational inertia and speed of a rotating body, analogous to linear momentum in translational motion.

The force causing rotation
The measure of rotational inertia and angular velocity
The resistance to linear motion
The acceleration of rotation
2

Two vectors are \( \mathbf{a} = 5 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = -3 \, \hat{\mathbf{i}} + 4 \, \hat{\mathbf{j}} \). What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?

\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 5 & 2 & 0 \\ -3 & 4 & 0 \end{vmatrix} = \hat{\mathbf{k}} (5 \times 4 - 2 \times (-3)) = \hat{\mathbf{k}} (20 + 6) = 26 \, \hat{\mathbf{k}} \).

Magnitude = \( 26 \).

24
25
26
27
3

In a rigid body, why does the point of application of force affect torque?

Torque depends on the perpendicular distance from the axis to the line of action of the force (\( \tau = r F \sin \theta \)), so the point of application changes \( r \), altering the torque.

Because force magnitude changes
Because perpendicular distance from the axis changes
Because angular velocity changes
Because mass distribution changes
2

Four particles of masses \( 1 \, \text{kg} \), \( 3 \, \text{kg} \), \( 5 \, \text{kg} \), and \( 7 \, \text{kg} \) are at \( (0, 0) \), \( (2, 0) \), \( (0, 3) \), and \( (2, 3) \) respectively. What is the y-coordinate of their center of mass?

Formula: \( Y = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{m_1 + m_2 + m_3 + m_4} \).

Masses: \( 1, 3, 5, 7 \, \text{kg} \); y-coordinates: \( 0, 0, 3, 3 \).

\( Y = \frac{(1 \times 0) + (3 \times 0) + (5 \times 3) + (7 \times 3)}{1 + 3 + 5 + 7} = \frac{0 + 0 + 15 + 21}{16} = \frac{36}{16} = 2.25 \, \text{m} \).

2.0 m
2.25 m
2.5 m
2.75 m
2

Why can a rigid body fixed at one point only undergo rotational motion?

When a rigid body is fixed at one point, translational motion is prevented because the fixed point cannot move, leaving rotation about that point as the only possible motion.

Because it has no mass
Because translational motion is impossible when fixed
Because it lacks angular velocity
Because it deforms under forces
2

A wheel with moment of inertia \( 4 \, \text{kg m}^2 \) rotates at \( 3 \, \text{rad/s} \). A torque of \( 12 \, \text{Nm} \) acts for \( 2 \, \text{s} \). What is its final angular velocity?

\( \alpha = \frac{\tau}{I} = \frac{12}{4} = 3 \, \text{rad/s}^2 \).

\( \Delta \omega = \alpha t = 3 \times 2 = 6 \, \text{rad/s} \).

\( \omega = \omega_0 + \Delta \omega = 3 + 6 = 9 \, \text{rad/s} \).

7 rad/s
9 rad/s
11 rad/s
13 rad/s
2

A uniform disk of mass \( 2 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rotates about its center. What is its moment of inertia?

For a uniform disk: \( I = \frac{1}{2} M R^2 \).

\( M = 2 \, \text{kg} \), \( R = 0.3 \, \text{m} \).

\( I = \frac{1}{2} \times 2 \times (0.3)^2 = 1 \times 0.09 = 0.09 \, \text{kg m}^2 \).

0.08 kg m²
0.09 kg m²
0.10 kg m²
0.12 kg m²
2

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