System of Particles and Rotational Motion Chapter-Wise Test 8

Correct answer Carries: 4.

Wrong Answer Carries: -1.

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

\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 2 & 3 & 0 \\ -3 & 4 & 0 \end{vmatrix} = \hat{\mathbf{k}} (2 \times 4 - 3 \times (-3)) = \hat{\mathbf{k}} (8 + 9) = 17 \, \hat{\mathbf{k}} \).

Magnitude = \( |\mathbf{a} \times \mathbf{b}| = 17 \).

15
16
17
18
3

What is the key assumption about distances in a rigid body during rotational motion?

A rigid body assumes that the distances between all pairs of particles remain constant, allowing it to rotate without deformation, which simplifies the analysis of rotational motion.

Distances between particles increase
Distances between particles remain constant
Distances between particles decrease
Distances vary randomly
2

A uniform disk of mass \( 3 \, \text{kg} \) and radius \( 0.6 \, \text{m} \) rotates about its center. What is its moment of inertia?

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

\( M = 3 \, \text{kg} \), \( R = 0.6 \, \text{m} \).

\( I = \frac{1}{2} \times 3 \times (0.6)^2 = 1.5 \times 0.36 = 0.54 \, \text{kg m}^2 \).

0.5 kg m²
0.54 kg m²
0.6 kg m²
0.7 kg m²
2

What does the moment of inertia of a rigid body depend on?

Moment of inertia (\( I = \sum m_i r_i^2 \)) depends on the mass of the body and the distribution of that mass relative to the axis of rotation.

Only the mass of the body
Only the shape of the body
Mass and its distribution about the axis
Only the angular velocity
3

Why does a spinning top precess when tilted?

Precession occurs because the torque due to gravity (acting on the top’s mass) causes a change in angular momentum direction, not magnitude, when the axis is not fixed, leading to a circular motion of the axis.

Due to increased angular velocity
Due to torque changing the direction of angular momentum
Due to zero moment of inertia
Due to linear acceleration
2

A hollow cylinder of mass \( 3 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates at \( 15 \, \text{rad/s} \). What is its angular momentum about its axis?

\( I = M R^2 = 3 \times (0.2)^2 = 0.12 \, \text{kg m}^2 \).

\( L = I \omega = 0.12 \times 15 = 1.8 \, \text{kg m}^2/\text{s} \).

1.5 kg m²/s
1.8 kg m²/s
2.0 kg m²/s
2.2 kg m²/s
2

Which statement is true about the vector product of two vectors?

The vector product (cross product) of two vectors results in a vector perpendicular to the plane of the two vectors, with magnitude \( ab \sin \theta \), and is not commutative (\( \mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a} \)).

It is commutative
It lies in the plane of the two vectors
It is perpendicular to the plane of the two vectors
Its magnitude is \( ab \cos \theta \)
3

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

For a hollow sphere: \( I = \frac{2}{3} M R^2 = \frac{2}{3} \times 2 \times (0.3)^2 = 0.12 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{6}{0.12} = 50 \, \text{rad/s} \).

40 rad/s
45 rad/s
50 rad/s
55 rad/s
3

What is the relationship between torque and angular acceleration for a rigid body?

Torque is directly proportional to angular acceleration (\( \mathbf{\tau} = I \mathbf{\alpha} \)), with moment of inertia as the constant of proportionality.

Inversely proportional
Directly proportional
Equal in magnitude
Independent of each other
2

A uniform rod of mass \( 6 \, \text{kg} \) and length \( 3 \, \text{m} \) is pivoted at one end. What is its moment of inertia about the pivot?

For a rod pivoted at one end: \( I = \frac{1}{3} M L^2 \).

\( M = 6 \, \text{kg} \), \( L = 3 \, \text{m} \).

\( I = \frac{1}{3} \times 6 \times (3)^2 = 2 \times 9 = 18 \, \text{kg m}^2 \).

15 kg m²
16 kg m²
18 kg m²
20 kg m²
3

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