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  4. System Of Particles And Rotational Motion Chapter-Wise Test 3

System Of Particles And Rotational Motion Chapter-Wise Test 3

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A wheel with moment of inertia \( 6 \, \text{kg m}^2 \) rotates at \( 3 \, \text{rad/s} \). A torque of \( 12 \, \text{Nm} \) acts for \( 3 \, \text{s} \). What is its final angular velocity?

\( \alpha = \frac{\tau}{I} = \frac{12}{6} = 2 \, \text{rad/s}^2 \).

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

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

7 rad/s
8 rad/s
9 rad/s
10 rad/s
3

A \( 3 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 6 \, \hat{\mathbf{i}} \, \text{m/s} \) at \( \mathbf{r} = 2 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 0 & 2 & 0 \\ 6 & 0 & 0 \end{vmatrix} = \hat{\mathbf{k}} (0 \times 0 - 2 \times 6) = -12 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

Magnitude = \( 12 \, \text{kg m}^2/\text{s} \).

10 kg m²/s
11 kg m²/s
12 kg m²/s
13 kg m²/s
3

A uniform triangular lamina has vertices at \( (0, 0) \), \( (5, 0) \), and \( (0, 6) \). What is the x-coordinate of its center of mass?

For a uniform triangular lamina, the center of mass is at the centroid, the average of the vertices.

\( X = \frac{0 + 5 + 0}{3} = \frac{5}{3} \approx 1.67 \, \text{m} \).

1.5 m
1.67 m
1.8 m
2.0 m
2

Vectors \( \mathbf{a} = 8 \, \hat{\mathbf{i}} - 3 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = 2 \, \hat{\mathbf{i}} + 7 \, \hat{\mathbf{j}} \) are given. What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?

\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 8 & -3 & 0 \\ 2 & 7 & 0 \end{vmatrix} = \hat{\mathbf{k}} (8 \times 7 - (-3) \times 2) = \hat{\mathbf{k}} (56 + 6) = 62 \, \hat{\mathbf{k}} \).

Magnitude = \( 62 \).

60
61
62
63
3

What is the primary source of angular acceleration in a rigid body?

Angular acceleration (\( \alpha \)) is caused by net external torque (\( \mathbf{\tau} = I \mathbf{\alpha} \)), which drives rotational change.

Linear force
Net external torque
Angular velocity
Mass of the body
2

A \( 7 \, \text{kg} \) object moves with a velocity of \( 3 \, \hat{\mathbf{i}} - 4 \, \hat{\mathbf{j}} \, \text{m/s} \). 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 = \( \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \).

4.5 m/s
5.0 m/s
5.5 m/s
6.0 m/s
2

A uniform rod of mass \( 2 \, \text{kg} \) and length \( 1 \, \text{m} \) is pivoted at one end. What is its moment of inertia about the pivot?

Moment of inertia of a rod about an end: \( I = \frac{1}{3} M L^2 \).

\( M = 2 \, \text{kg} \), \( L = 1 \, \text{m} \).

\( I = \frac{1}{3} \times 2 \times (1)^2 = \frac{2}{3} \approx 0.67 \, \text{kg m}^2 \).

0.33 kg m²
0.5 kg m²
0.67 kg m²
1 kg m²
3

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

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

Magnitude = \( 26 \).

24
25
26
27
3

A hollow cylinder of mass \( 1 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) has a kinetic energy of \( 2 \, \text{J} \). What is its angular speed?

\( I = M R^2 = 1 \times (0.2)^2 = 0.04 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 \Rightarrow 2 = \frac{1}{2} \times 0.04 \times \omega^2 \Rightarrow 4 = 0.04 \omega^2 \Rightarrow \omega^2 = 100 \Rightarrow \omega = 10 \, \text{rad/s} \).

8 rad/s
10 rad/s
12 rad/s
14 rad/s
2

What characterizes pure translational motion of a rigid body?

In pure translational motion, all particles of the rigid body have the same velocity at any given instant, with no rotation involved.

All particles have the same angular velocity
All particles have the same linear velocity
Particles rotate about a fixed axis
Particles have different velocities depending on their position
2

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