System of Particles and Rotational Motion Chapter-Wise Test 1

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A \( 5 \, \text{kg} \) object moves with a velocity of \( -3 \, \hat{\mathbf{i}} + 6 \, \hat{\mathbf{j}} \, \text{m/s} \). 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 = \( \sqrt{(-3)^2 + (6)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71 \, \text{m/s} \).

6.5 m/s
6.71 m/s
6.9 m/s
7.0 m/s
2

What is the primary difference between translational and rotational motion of a rigid body?

In translational motion, all particles move with the same linear velocity, while in rotational motion, particles move in circular paths with the same angular velocity about a fixed axis.

Translational motion involves rotation about an axis
Rotational motion involves all particles moving with the same linear velocity
Translational motion involves all particles moving with the same velocity, while rotational motion involves the same angular velocity
Rotational motion occurs without an axis
3

A hollow cylinder of mass \( 3 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rotates about its axis with an angular speed of \( 6 \, \text{rad/s} \). What is its rotational kinetic energy?

For a hollow cylinder: \( I = M R^2 = 3 \times (0.3)^2 = 0.27 \, \text{kg m}^2 \).

KE: \( K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.27 \times (6)^2 = 0.135 \times 36 = 4.86 \, \text{J} \approx 4.9 \, \text{J} \).

4.5 J
4.9 J
5.2 J
5.5 J
2

What does torque represent in rotational motion?

Torque is the rotational equivalent of force, causing angular acceleration. It’s defined as \( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \), with magnitude \( r F \sin \theta \).

The linear force applied to a body
The rotational effect of a force about an axis
The mass distribution of a body
The velocity of rotation
2

A thin ring of mass \( 4 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) rolls without slipping with a linear speed of \( 5 \, \text{m/s} \). What is its angular velocity?

\( v = \omega r \).

\( v = 5 \, \text{m/s} \), \( r = 0.5 \, \text{m} \).

\( \omega = \frac{v}{r} = \frac{5}{0.5} = 10 \, \text{rad/s} \).

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

A hollow cylinder of mass \( 3 \, \text{kg} \) and radius \( 0.8 \, \text{m} \) rolls without slipping with an angular velocity of \( 6 \, \text{rad/s} \). What is its linear speed?

\( v = \omega r \).

\( \omega = 6 \, \text{rad/s} \), \( r = 0.8 \, \text{m} \).

\( v = 6 \times 0.8 = 4.8 \, \text{m/s} \).

4.0 m/s
4.5 m/s
4.8 m/s
5.0 m/s
3

Why is the vector product of two parallel vectors zero?

The vector product \( \mathbf{a} \times \mathbf{b} = ab \sin \theta \) is zero for parallel vectors because \( \theta = 0^\circ \) or \( 180^\circ \), making \( \sin \theta = 0 \).

Because their magnitudes are zero
Because the angle between them is zero or 180°
Because they lie in different planes
Because their directions are opposite
2

In pure rotational motion about a fixed axis, what is true about the motion of particles in a rigid body?

In pure rotational motion about a fixed axis, every particle moves in a circular path in a plane perpendicular to the axis, with the center of the circle on the axis.

All particles move in straight lines
All particles move in circles with centers on the axis
Particles move randomly in all directions
Only particles on the axis move in circles
2

A \( 5 \, \text{kg} \) mass rotates in a circle of radius \( 0.4 \, \text{m} \) with a speed of \( 3 \, \text{m/s} \). What is its angular momentum about the center?

\( L = m v r \).

\( m = 5 \, \text{kg} \), \( v = 3 \, \text{m/s} \), \( r = 0.4 \, \text{m} \).

\( L = 5 \times 3 \times 0.4 = 6 \, \text{kg m}^2/\text{s} \).

5.5 kg m²/s
6.0 kg m²/s
6.5 kg m²/s
7.0 kg m²/s
2

A thin ring of mass \( 3 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) has an angular momentum of \( 5.4 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = M R^2 = 3 \times (0.3)^2 = 0.27 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{5.4}{0.27} = 20 \, \text{rad/s} \).

15 rad/s
18 rad/s
20 rad/s
22 rad/s
3

A \( 2 \, \text{kg} \) mass is attached to a \( 1 \, \text{m} \) long string and rotates in a horizontal circle. If its linear speed is \( 3 \, \text{m/s} \), what is its angular momentum about the center?

Angular momentum: \( L = m v r \).

\( m = 2 \, \text{kg} \), \( v = 3 \, \text{m/s} \), \( r = 1 \, \text{m} \).

\( L = 2 \times 3 \times 1 = 6 \, \text{kg m}^2/\text{s} \).

4 kg m²/s
5 kg m²/s
6 kg m²/s
7 kg m²/s
3

A \( 3 \, \text{kg} \) block moves with a velocity of \( 5 \, \text{m/s} \) along the y-axis. What is the magnitude of the velocity of its center of mass?

For a single object, the velocity of the center of mass is the same as the object’s velocity.

Velocity = \( 5 \, \text{m/s} \) along y-axis, magnitude = \( 5 \, \text{m/s} \).

3 m/s
4 m/s
5 m/s
6 m/s
3

In rotational kinematics, what is the rotational analogue of linear acceleration?

Angular acceleration (\( \alpha \)) is the rotational analogue of linear acceleration, representing the rate of change of angular velocity (\( \alpha = \frac{d\omega}{dt} \)).

Angular velocity
Angular acceleration
Torque
Moment of inertia
2

Vectors \( \mathbf{a} = 2 \, \hat{\mathbf{i}} + 6 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = 5 \, \hat{\mathbf{i}} - 3 \, \hat{\mathbf{j}} \) are given. What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?

\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 2 & 6 & 0 \\ 5 & -3 & 0 \end{vmatrix} = \hat{\mathbf{k}} (2 \times (-3) - 6 \times 5) = \hat{\mathbf{k}} (-6 - 30) = -36 \, \hat{\mathbf{k}} \).

Magnitude = \( 36 \).

34
35
36
37
3

What is the unit of torque in the SI system?

Torque is the product of force (N) and distance (m), so its SI unit is Newton-meter (Nm).

kg m/s²
N m
kg m²/s²
rad/s²
2

Performance Summary

Score:

Category Details
Total Attempts:
Total Skipped:
Total Wrong Answers:
Total Correct Answers:
Time Taken:
Average Time Taken per Question:
Accuracy:
0