System of Particles and Rotational Motion Chapter-Wise Test 9

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A solid cylinder of mass \( 4 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rotates about its axis at \( 6 \, \text{rad/s} \). What is its rotational kinetic energy?

\( I = \frac{1}{2} M R^2 = \frac{1}{2} \times 4 \times (0.3)^2 = 0.18 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.18 \times (6)^2 = 0.09 \times 36 = 3.24 \, \text{J} \).

3.0 J
3.24 J
3.5 J
4.0 J
2

What happens to the rotational kinetic energy if the angular velocity of a body is halved?

Rotational kinetic energy \( K = \frac{1}{2} I \omega^2 \) depends on \( \omega^2 \). Halving \( \omega \) reduces \( \omega^2 \) to \( \frac{1}{4} \), so \( K \) becomes one-fourth its original value.

It doubles
It halves
It becomes one-fourth
It remains unchanged
3

A torque of \( 20 \, \text{Nm} \) acts on a wheel with moment of inertia \( 10 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 4 \, \text{s} \)?

\( \alpha = \frac{\tau}{I} = \frac{20}{10} = 2 \, \text{rad/s}^2 \).

\( \omega = \omega_0 + \alpha t = 0 + 2 \times 4 = 8 \, \text{rad/s} \).

6 rad/s
8 rad/s
10 rad/s
12 rad/s
2

A thin ring of mass \( 2 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has a kinetic energy of \( 8 \, \text{J} \). What is its angular speed?

\( I = M R^2 = 2 \times (0.4)^2 = 0.32 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 \Rightarrow 8 = \frac{1}{2} \times 0.32 \times \omega^2 \Rightarrow 16 = 0.32 \omega^2 \Rightarrow \omega^2 = 50 \Rightarrow \omega = \sqrt{50} \approx 7.07 \, \text{rad/s} \).

6 rad/s
7 rad/s
8 rad/s
9 rad/s
2

A thin ring of mass \( 2 \, \text{kg} \) and radius \( 0.7 \, \text{m} \) rolls without slipping with a linear speed of \( 3.5 \, \text{m/s} \). What is its angular velocity?

\( v = \omega r \).

\( v = 3.5 \, \text{m/s} \), \( r = 0.7 \, \text{m} \).

\( \omega = \frac{v}{r} = \frac{3.5}{0.7} = 5 \, \text{rad/s} \).

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

A \( 1 \, \text{kg} \) mass rotates in a circle of radius \( 0.8 \, \text{m} \) with a speed of \( 5 \, \text{m/s} \). What is its angular momentum about the center?

\( L = m v r \).

\( m = 1 \, \text{kg} \), \( v = 5 \, \text{m/s} \), \( r = 0.8 \, \text{m} \).

\( L = 1 \times 5 \times 0.8 = 4 \, \text{kg m}^2/\text{s} \).

3.5 kg m²/s
4.0 kg m²/s
4.5 kg m²/s
5.0 kg m²/s
2

A solid cylinder of mass \( 5 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) has an angular momentum of \( 12.5 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = \frac{1}{2} M R^2 = \frac{1}{2} \times 5 \times (0.5)^2 = 0.625 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{12.5}{0.625} = 20 \, \text{rad/s} \).

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

A solid cylinder of mass \( 2 \, \text{kg} \) and radius \( 0.1 \, \text{m} \) rotates about its axis at \( 20 \, \text{rad/s} \). What is its rotational kinetic energy?

\( I = \frac{1}{2} M R^2 = \frac{1}{2} \times 2 \times (0.1)^2 = 0.01 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.01 \times (20)^2 = 0.005 \times 400 = 2 \, \text{J} \).

1.5 J
2.0 J
2.5 J
3.0 J
2

A torque of \( 18 \, \text{Nm} \) is applied to a wheel with moment of inertia \( 9 \, \text{kg m}^2 \). What is its angular acceleration?

\( \tau = I \alpha \).

\( \tau = 18 \, \text{Nm} \), \( I = 9 \, \text{kg m}^2 \).

\( \alpha = \frac{\tau}{I} = \frac{18}{9} = 2 \, \text{rad/s}^2 \).

1.5 rad/s²
2.0 rad/s²
2.5 rad/s²
3.0 rad/s²
2

A particle rotates about a fixed axis at a radius of \( 0.3 \, \text{m} \) with a linear speed of \( 6 \, \text{m/s} \). What is its angular velocity?

\( v = \omega r \).

\( v = 6 \, \text{m/s} \), \( r = 0.3 \, \text{m} \).

\( \omega = \frac{v}{r} = \frac{6}{0.3} = 20 \, \text{rad/s} \).

15 rad/s
18 rad/s
20 rad/s
25 rad/s
3

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