System of Particles and Rotational Motion Chapter-Wise Test 2

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A uniform rectangular lamina of dimensions \( 2 \, \text{m} \times 4 \, \text{m} \) and mass \( 10 \, \text{kg} \) has one corner at the origin along the x- and y-axes. What is the position of its center of mass?

For a uniform rectangle, the center of mass is at the centroid.

Vertices: \( (0, 0), (2, 0), (0, 4), (2, 4) \).

CM: \( X = \frac{0 + 2}{2} = 1 \), \( Y = \frac{0 + 4}{2} = 2 \). Position: \( (1, 2) \, \text{m} \).

\( (1, 1) \)
\( (1, 2) \)
\( (2, 2) \)
\( (2, 3) \)
2

A \( 4 \, \text{kg} \) mass rotates in a circle of radius \( 0.25 \, \text{m} \) with a linear speed of \( 2 \, \text{m/s} \). What is its angular momentum about the center?

\( L = m v r \).

\( m = 4 \, \text{kg} \), \( v = 2 \, \text{m/s} \), \( r = 0.25 \, \text{m} \).

\( L = 4 \times 2 \times 0.25 = 2 \, \text{kg m}^2/\text{s} \).

1.5 kg m²/s
2 kg m²/s
2.5 kg m²/s
3 kg m²/s
2

A thin uniform ring of mass \( 2 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) rotates about its center. What is its moment of inertia?

For a thin ring: \( I = M R^2 \).

\( M = 2 \, \text{kg} \), \( R = 0.5 \, \text{m} \).

\( I = 2 \times (0.5)^2 = 0.5 \, \text{kg m}^2 \).

0.25 kg m²
0.5 kg m²
0.75 kg m²
1.0 kg m²
2

A \( 2 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 5 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 3 \, \hat{\mathbf{i}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 3 & 0 & 0 \\ 0 & 5 & 0 \end{vmatrix} = \hat{\mathbf{k}} (3 \times 5 - 0 \times 0) = 15 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

Magnitude = \( 15 \, \text{kg m}^2/\text{s} \).

12 kg m²/s
13 kg m²/s
15 kg m²/s
17 kg m²/s
3

A particle in a rigid body rotates at a distance of \( 0.8 \, \text{m} \) from a fixed axis with an angular velocity of \( 7 \, \text{rad/s} \). What is its linear speed?

\( v = \omega r \).

\( \omega = 7 \, \text{rad/s} \), \( r = 0.8 \, \text{m} \).

\( v = 7 \times 0.8 = 5.6 \, \text{m/s} \).

5.0 m/s
5.6 m/s
6.0 m/s
6.4 m/s
2

Three particles of masses \( 2 \, \text{kg} \), \( 4 \, \text{kg} \), and \( 8 \, \text{kg} \) are at \( (0, 3) \), \( (4, 0) \), and \( (2, 5) \) respectively. What is the y-coordinate of their center of mass?

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

Masses: \( 2, 4, 8 \, \text{kg} \); y-coordinates: \( 3, 0, 5 \).

\( Y = \frac{(2 \times 3) + (4 \times 0) + (8 \times 5)}{2 + 4 + 8} = \frac{6 + 0 + 40}{14} = \frac{46}{14} \approx 3.29 \, \text{m} \).

3.0 m
3.29 m
3.5 m
3.7 m
2

A wheel with moment of inertia \( 2 \, \text{kg m}^2 \) rotates at \( 4 \, \text{rad/s} \). A torque of \( 6 \, \text{Nm} \) acts for \( 4 \, \text{s} \). What is its final angular velocity?

\( \alpha = \frac{\tau}{I} = \frac{6}{2} = 3 \, \text{rad/s}^2 \).

\( \Delta \omega = \alpha t = 3 \times 4 = 12 \, \text{rad/s} \).

\( \omega = \omega_0 + \Delta \omega = 4 + 12 = 16 \, \text{rad/s} \).

14 rad/s
15 rad/s
16 rad/s
17 rad/s
3

What condition indicates that a rigid body is not rotating?

A rigid body is not rotating if its angular velocity is zero (\( \omega = 0 \)), meaning there’s no change in angular position over time.

Net torque is zero
Angular velocity is zero
Moment of inertia is zero
Linear velocity is zero
2

A thin ring of mass \( 6 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has an angular momentum of \( 9.6 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = M R^2 = 6 \times (0.4)^2 = 0.96 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{9.6}{0.96} = 10 \, \text{rad/s} \).

9 rad/s
10 rad/s
11 rad/s
12 rad/s
2

A solid sphere of mass \( 1 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates about its center at \( 10 \, \text{rad/s} \). What is its rotational kinetic energy?

\( I = \frac{2}{5} M R^2 = \frac{2}{5} \times 1 \times (0.2)^2 = 0.016 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.016 \times (10)^2 = 0.008 \times 100 = 0.8 \, \text{J} \).

0.6 J
0.8 J
1.0 J
1.2 J
2

A \( 6 \, \text{kg} \) object moves with a velocity of \( 2 \, \hat{\mathbf{i}} + 5 \, \hat{\mathbf{j}} \, \text{m/s} \). What is the magnitude of the velocity of its center of mass?

For a single object, the center of mass velocity equals the object’s velocity.

Magnitude = \( \sqrt{(2)^2 + (5)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.39 \, \text{m/s} \).

5.0 m/s
5.39 m/s
5.6 m/s
6.0 m/s
2

A thin ring of mass \( 5 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has an angular momentum of \( 8 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = M R^2 = 5 \times (0.4)^2 = 0.8 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{8}{0.8} = 10 \, \text{rad/s} \).

8 rad/s
9 rad/s
10 rad/s
11 rad/s
3

What happens to the angular momentum of a rigid body if its moment of inertia increases while no external torque acts?

If no external torque acts (\( \mathbf{\tau} = 0 \)), angular momentum (\( \mathbf{L} = I \mathbf{\omega} \)) is conserved. If \( I \) increases, \( \omega \) must decrease to keep \( L \) constant.

It increases
It decreases
It remains constant
It becomes zero
3

A thin ring of mass \( 2 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) has a kinetic energy of \( 10 \, \text{J} \). What is its angular speed?

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

\( K = \frac{1}{2} I \omega^2 \Rightarrow 10 = \frac{1}{2} \times 0.5 \times \omega^2 \Rightarrow 20 = 0.5 \omega^2 \Rightarrow \omega^2 = 40 \Rightarrow \omega = \sqrt{40} \approx 6.32 \, \text{rad/s} \).

6.0 rad/s
6.32 rad/s
6.5 rad/s
7.0 rad/s
2

What is the significance of the perpendicular component of force in calculating torque?

Torque (\( \tau = r F \sin \theta \)) depends on the perpendicular component of force (\( F \sin \theta \)), as only this component causes rotation about the pivot, not the radial component.

It determines linear acceleration
It causes rotational effect
It balances the body
It affects mass distribution
2

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