System of Particles and Rotational Motion Chapter-Wise Test 17

Correct answer Carries: 4.

Wrong Answer Carries: -1.

Three particles of masses \( 3 \, \text{kg} \), \( 6 \, \text{kg} \), and \( 9 \, \text{kg} \) are located at \( (0, 1) \), \( (2, 2) \), and \( (4, 0) \) respectively. What is the x-coordinate of their center of mass?

Formula: \( X = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} \).

Masses: \( 3, 6, 9 \, \text{kg} \); x-coordinates: \( 0, 2, 4 \).

\( X = \frac{(3 \times 0) + (6 \times 2) + (9 \times 4)}{3 + 6 + 9} = \frac{0 + 12 + 36}{18} = \frac{48}{18} = 2.67 \, \text{m} \).

2.5 m
2.67 m
2.8 m
3.0 m
2

A solid sphere 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?

\( I = \frac{2}{5} M R^2 = \frac{2}{5} \times 3 \times (0.4)^2 = 0.192 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.192 \times (5)^2 = 0.096 \times 25 = 2.4 \, \text{J} \).

2.0 J
2.4 J
2.8 J
3.0 J
2

A torque of \( 14 \, \text{Nm} \) acts on a wheel with moment of inertia \( 7 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 5 \, \text{s} \)?

\( \alpha = \frac{\tau}{I} = \frac{14}{7} = 2 \, \text{rad/s}^2 \).

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

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

A \( 5 \, \text{kg} \) block slides down an inclined plane with a constant velocity of \( 2 \, \text{m/s} \). What is the velocity of the center of mass of the block?

For a single body in pure translational motion, the velocity of the center of mass is the same as the velocity of the body.

Given: \( v = 2 \, \text{m/s} \).

Thus, \( \mathbf{V} = 2 \, \text{m/s} \) (direction along the incline).

1 m/s
2 m/s
3 m/s
4 m/s
2

What is the condition for a rigid body to have zero angular acceleration?

Angular acceleration is zero when the net torque is zero (\( \mathbf{\tau} = I \mathbf{\alpha} \)), as torque drives rotational change.

Net force is zero
Net torque is zero
Angular velocity is maximum
Moment of inertia is infinite
2

A torque of \( 15 \, \text{Nm} \) is applied to a wheel with moment of inertia \( 3 \, \text{kg m}^2 \). What is its angular acceleration?

\( \tau = I \alpha \).

\( \tau = 15 \, \text{Nm} \), \( I = 3 \, \text{kg m}^2 \).

\( \alpha = \frac{\tau}{I} = \frac{15}{3} = 5 \, \text{rad/s}^2 \).

4 rad/s²
5 rad/s²
6 rad/s²
7 rad/s²
2

A particle rotates about a fixed axis at a radius of \( 0.6 \, \text{m} \) with a linear speed of \( 3 \, \text{m/s} \). What is its angular velocity?

\( v = \omega r \).

\( v = 3 \, \text{m/s} \), \( r = 0.6 \, \text{m} \).

\( \omega = \frac{v}{r} = \frac{3}{0.6} = 5 \, \text{rad/s} \).

4 rad/s
5 rad/s
6 rad/s
7 rad/s
2

Three particles of masses \( 1 \, \text{kg} \), \( 5 \, \text{kg} \), and \( 10 \, \text{kg} \) are at \( (0, 2) \), \( (3, 0) \), and \( (4, 6) \) respectively. What is the x-coordinate of their center of mass?

Formula: \( X = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} \).

Masses: \( 1, 5, 10 \, \text{kg} \); x-coordinates: \( 0, 3, 4 \).

\( X = \frac{(1 \times 0) + (5 \times 3) + (10 \times 4)}{1 + 5 + 10} = \frac{0 + 15 + 40}{16} = \frac{55}{16} \approx 3.44 \, \text{m} \).

3.2 m
3.44 m
3.6 m
3.8 m
2

Which physical quantity is defined as \( \mathbf{r} \times \mathbf{p} \) for a particle?

Angular momentum of a particle is defined as \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \), where \( \mathbf{r} \) is the position vector and \( \mathbf{p} \) is the linear momentum.

Torque
Angular velocity
Angular momentum
Moment of inertia
3

Why can a rigid body not deform under the influence of forces in theoretical mechanics?

In theoretical mechanics, a rigid body is an idealization where inter-particle distances are fixed, preventing deformation and allowing focus on rotational and translational motion only.

Because it has infinite mass
Because it is an idealization with fixed distances
Because it lacks rotational motion
Because forces are always zero
2

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