System of Particles and Rotational Motion Chapter-Wise Test 6

Correct answer Carries: 4.

Wrong Answer Carries: -1.

Why does a dancer spin faster when pulling their arms closer to their body?

Pulling arms closer reduces the moment of inertia (\( I \)). With no external torque, angular momentum (\( L = I \omega \)) is conserved, so \( \omega \) increases as \( I \) decreases.

Due to increased torque
Due to conservation of angular momentum
Due to increased linear velocity
Due to decreased mass
2

A force \( \mathbf{F} = 5 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \, \text{N} \) acts at \( \mathbf{r} = -1 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \, \text{m} \). What is the torque about the origin?

\( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ -1 & 3 & 0 \\ 5 & 2 & 0 \end{vmatrix} = \hat{\mathbf{k}} ((-1) \times 2 - 3 \times 5) = \hat{\mathbf{k}} (-2 - 15) = -17 \, \hat{\mathbf{k}} \, \text{Nm} \).

-19 Nm
-17 Nm
-15 Nm
-13 Nm
2

A torque of \( 24 \, \text{Nm} \) acts on a wheel with moment of inertia \( 6 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 3 \, \text{s} \)?

\( \alpha = \frac{\tau}{I} = \frac{24}{6} = 4 \, \text{rad/s}^2 \).

\( \omega = \omega_0 + \alpha t = 0 + 4 \times 3 = 12 \, \text{rad/s} \).

10 rad/s
11 rad/s
12 rad/s
13 rad/s
3

A particle moves in a circle of radius \( 0.5 \, \text{m} \) with an angular velocity of \( 4 \, \text{rad/s} \). What is the magnitude of its linear velocity?

Linear velocity: \( v = \omega r \).

\( \omega = 4 \, \text{rad/s} \), \( r = 0.5 \, \text{m} \).

\( v = 4 \times 0.5 = 2 \, \text{m/s} \).

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

A particle rotates about a fixed axis at a radius of \( 0.6 \, \text{m} \) with a linear speed of \( 7.2 \, \text{m/s} \). What is its angular velocity?

\( v = \omega r \).

\( v = 7.2 \, \text{m/s} \), \( r = 0.6 \, \text{m} \).

\( \omega = \frac{v}{r} = \frac{7.2}{0.6} = 12 \, \text{rad/s} \).

10 rad/s
11 rad/s
12 rad/s
13 rad/s
3

A \( 5 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 2 \, \hat{\mathbf{i}} - 3 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 4 \, \hat{\mathbf{j}} \, \text{m} \). What is the z-component of its angular momentum?

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 0 & 4 & 0 \\ 2 & -3 & 0 \end{vmatrix} = \hat{\mathbf{k}} (0 \times (-3) - 4 \times 2) = -8 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

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

-10 kg m²/s
-8 kg m²/s
-6 kg m²/s
-4 kg m²/s
2

What is the effect of increasing the mass of a rotating body on its angular momentum, if angular velocity remains constant?

Angular momentum \( \mathbf{L} = I \mathbf{\omega} \), and \( I \) increases with mass. If \( \omega \) is constant, \( L \) increases proportionally with mass.

It decreases
It remains constant
It increases
It becomes zero
3

A solid sphere of mass \( 2 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rolls without slipping with an angular velocity of \( 10 \, \text{rad/s} \). What is its linear speed?

\( v = \omega r \).

\( \omega = 10 \, \text{rad/s} \), \( r = 0.3 \, \text{m} \).

\( v = 10 \times 0.3 = 3 \, \text{m/s} \).

2.5 m/s
3.0 m/s
3.5 m/s
4.0 m/s
2

A uniform rod of mass \( 3 \, \text{kg} \) and length \( 1.5 \, \text{m} \) is pivoted at one end. What is its moment of inertia about the pivot?

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

\( M = 3 \, \text{kg} \), \( L = 1.5 \, \text{m} \).

\( I = \frac{1}{3} \times 3 \times (1.5)^2 = 1 \times 2.25 = 2.25 \, \text{kg m}^2 \).

2.0 kg m²
2.25 kg m²
2.5 kg m²
2.75 kg m²
2

A particle in a rigid body rotates at a distance of \( 0.7 \, \text{m} \) from a fixed axis with an angular velocity of \( 6 \, \text{rad/s} \). What is its linear speed?

\( v = \omega r \).

\( \omega = 6 \, \text{rad/s} \), \( r = 0.7 \, \text{m} \).

\( v = 6 \times 0.7 = 4.2 \, \text{m/s} \).

3.8 m/s
4.0 m/s
4.2 m/s
4.5 m/s
3

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

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 6 & 0 & 0 \\ 0 & -4 & 0 \end{vmatrix} = \hat{\mathbf{k}} (6 \times (-4) - 0 \times 0) = -24 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

Magnitude = \( 24 \, \text{kg m}^2/\text{s} \).

22 kg m²/s
23 kg m²/s
24 kg m²/s
25 kg m²/s
3

A particle rotates about a fixed axis at a radius of \( 0.9 \, \text{m} \) with a linear speed of \( 4.5 \, \text{m/s} \). What is its angular velocity?

\( v = \omega r \).

\( v = 4.5 \, \text{m/s} \), \( r = 0.9 \, \text{m} \).

\( \omega = \frac{v}{r} = \frac{4.5}{0.9} = 5 \, \text{rad/s} \).

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

A uniform rod of mass \( 5 \, \text{kg} \) and length \( 1.2 \, \text{m} \) is pivoted at its center. What is its moment of inertia about the pivot?

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

\( M = 5 \, \text{kg} \), \( L = 1.2 \, \text{m} \).

\( I = \frac{1}{12} \times 5 \times (1.2)^2 = \frac{5 \times 1.44}{12} = 0.6 \, \text{kg m}^2 \).

0.5 kg m²
0.6 kg m²
0.7 kg m²
0.8 kg m²
2

A torque of \( 12 \, \text{Nm} \) acts on a wheel with moment of inertia \( 3 \, \text{kg m}^2 \) for \( 4 \, \text{s} \). If it starts from rest, what is its final angular speed?

\( \tau = I \alpha \Rightarrow \alpha = \frac{\tau}{I} = \frac{12}{3} = 4 \, \text{rad/s}^2 \).

\( \omega = \omega_0 + \alpha t = 0 + 4 \times 4 = 16 \, \text{rad/s} \).

12 rad/s
14 rad/s
16 rad/s
18 rad/s
3

A torque of \( 20 \, \text{Nm} \) acts on a wheel with moment of inertia \( 8 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 5 \, \text{s} \)?

\( \alpha = \frac{\tau}{I} = \frac{20}{8} = 2.5 \, \text{rad/s}^2 \).

\( \omega = \omega_0 + \alpha t = 0 + 2.5 \times 5 = 12.5 \, \text{rad/s} \).

10 rad/s
12.5 rad/s
15 rad/s
17.5 rad/s
2

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