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  4. System Of Particles And Rotational Motion Chapter-Wise Test 2

System Of Particles And Rotational Motion Chapter-Wise Test 2

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A solid sphere of mass \( 6 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has an angular momentum of \( 11.52 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = \frac{2}{5} M R^2 = \frac{2}{5} \times 6 \times (0.4)^2 = 0.384 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{11.52}{0.384} = 30 \, \text{rad/s} \).

28 rad/s
30 rad/s
32 rad/s
34 rad/s
2

A uniform rod of mass \( 5 \, \text{kg} \) and length \( 2 \, \text{m} \) is pivoted at its center. What is its moment of inertia about the pivot?

For a rod pivoted at its center: \( I = \frac{1}{12} M L^2 \).

\( M = 5 \, \text{kg} \), \( L = 2 \, \text{m} \).

\( I = \frac{1}{12} \times 5 \times (2)^2 = \frac{20}{12} = \frac{5}{3} \approx 1.67 \, \text{kg m}^2 \).

1.5 kg m²
1.67 kg m²
2.0 kg m²
2.5 kg m²
2

In the expression \( \mathbf{v} = \mathbf{\omega} \times \mathbf{r} \), what does \( \mathbf{v} \) represent?

The equation \( \mathbf{v} = \mathbf{\omega} \times \mathbf{r} \) gives the linear velocity of a particle in rotational motion, where \( \mathbf{\omega} \) is the angular velocity and \( \mathbf{r} \) is the position vector from the axis.

Angular velocity
Linear velocity
Torque
Angular momentum
2

A thin ring of mass \( 3 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) rotates about its center at \( 5 \, \text{rad/s} \). What is its rotational kinetic energy?

\( I = M R^2 = 3 \times (0.4)^2 = 0.48 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.48 \times (5)^2 = 0.24 \times 25 = 6 \, \text{J} \).

5.5 J
6.0 J
6.5 J
7.0 J
2

What happens to the angular velocity of a rigid body if the net external torque is zero?

If the net external torque is zero (\( \frac{d\mathbf{L}}{dt} = \mathbf{\tau} = 0 \)), angular momentum is conserved, and for a fixed moment of inertia, angular velocity remains constant.

It increases
It decreases
It remains constant
It becomes zero
3

A thin ring of mass \( 1 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has a kinetic energy of \( 8 \, \text{J} \). What is its angular speed?

\( I = M R^2 = 1 \times (0.4)^2 = 0.16 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 \Rightarrow 8 = \frac{1}{2} \times 0.16 \times \omega^2 \Rightarrow 16 = 0.16 \omega^2 \Rightarrow \omega^2 = 100 \Rightarrow \omega = 10 \, \text{rad/s} \).

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

Which of the following is a requirement for a rigid body to be in mechanical equilibrium?

For mechanical equilibrium, both translational and rotational equilibrium are required: the net force (\( \sum \mathbf{F} = 0 \)) and net torque (\( \sum \mathbf{\tau} = 0 \)) must be zero.

Net force must be zero only
Net torque must be zero only
Both net force and net torque must be zero
Angular velocity must be constant
3

A \( 3 \, \text{kg} \) particle has a velocity \( \mathbf{v} = 2 \, \hat{\mathbf{i}} \, \text{m/s} \) at position \( \mathbf{r} = 4 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times (m \mathbf{v}) = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 0 & 4 & 0 \\ 2 & 0 & 0 \end{vmatrix} = -8 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

Magnitude = \( 8 \, \text{kg m}^2/\text{s} \).

6 kg m²/s
8 kg m²/s
10 kg m²/s
12 kg m²/s
2

A torque of \( 25 \, \text{Nm} \) acts on a wheel with moment of inertia \( 5 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 2 \, \text{s} \)?

\( \alpha = \frac{\tau}{I} = \frac{25}{5} = 5 \, \text{rad/s}^2 \).

\( \omega = \omega_0 + \alpha t = 0 + 5 \times 2 = 10 \, \text{rad/s} \).

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

A \( 3 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 6 \, \hat{\mathbf{i}} - 4 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 2 \, \hat{\mathbf{j}} \, \text{m} \). What is the z-component of its angular momentum?

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 0 & 2 & 0 \\ 6 & -4 & 0 \end{vmatrix} = \hat{\mathbf{k}} (0 \times (-4) - 2 \times 6) = -12 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

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

-14 kg m²/s
-12 kg m²/s
-10 kg m²/s
-8 kg m²/s
2

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