System of Particles and Rotational Motion Chapter-Wise Test 19

Correct answer Carries: 4.

Wrong Answer Carries: -1.

Two particles of masses \( 3 \, \text{kg} \) and \( 7 \, \text{kg} \) are at \( (1, 4) \) and \( (9, 2) \) respectively. What is the distance of their center of mass from the origin?

\( X = \frac{(3 \times 1) + (7 \times 9)}{3 + 7} = \frac{3 + 63}{10} = 6.6 \).

\( Y = \frac{(3 \times 4) + (7 \times 2)}{3 + 7} = \frac{12 + 14}{10} = 2.6 \).

Distance = \( \sqrt{(6.6)^2 + (2.6)^2} = \sqrt{43.56 + 6.76} = \sqrt{50.32} \approx 7.09 \, \text{m} \).

6.8 m
7.09 m
7.3 m
7.5 m
2

A solid cylinder of mass \( 2 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) has a kinetic energy of \( 10 \, \text{J} \). What is its angular speed?

\( I = \frac{1}{2} M R^2 = \frac{1}{2} \times 2 \times (0.5)^2 = 0.25 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 \Rightarrow 10 = \frac{1}{2} \times 0.25 \times \omega^2 \Rightarrow 20 = 0.25 \omega^2 \Rightarrow \omega^2 = 80 \Rightarrow \omega = \sqrt{80} \approx 8.94 \, \text{rad/s} \).

8.5 rad/s
8.94 rad/s
9.5 rad/s
10 rad/s
2

A hollow cylinder of mass \( 4 \, \text{kg} \) and radius \( 0.6 \, \text{m} \) rotates at \( 5 \, \text{rad/s} \). What is its angular momentum about its axis?

\( I = M R^2 = 4 \times (0.6)^2 = 1.44 \, \text{kg m}^2 \).

\( L = I \omega = 1.44 \times 5 = 7.2 \, \text{kg m}^2/\text{s} \).

6.8 kg m²/s
7.0 kg m²/s
7.2 kg m²/s
7.5 kg m²/s
3

A solid sphere of mass \( 5 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rotates about an axis through its center. If its angular momentum is \( 9 \, \text{kg m}^2/\text{s} \), what is its angular velocity?

\( L = I \omega \).

For a sphere: \( I = \frac{2}{5} M R^2 = \frac{2}{5} \times 5 \times (0.3)^2 = 0.18 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{9}{0.18} = 50 \, \text{rad/s} \).

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

A uniform rectangular plate of dimensions \( 3 \, \text{m} \times 5 \, \text{m} \) and mass \( 12 \, \text{kg} \) has one corner at \( (1, 1) \) along the x- and y-axes. What is the position of its center of mass?

For a uniform rectangle, the center of mass is at the centroid.

Vertices: \( (1, 1), (4, 1), (1, 6), (4, 6) \).

CM: \( X = 1 + \frac{3}{2} = 2.5 \), \( Y = 1 + \frac{5}{2} = 3.5 \). Position: \( (2.5, 3.5) \, \text{m} \).

\( (2, 3) \)
\( (2.5, 3) \)
\( (2.5, 3.5) \)
\( (3, 3.5) \)
3

A solid cylinder of mass \( 4 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates at \( 25 \, \text{rad/s} \). What is its angular momentum about its axis?

\( I = \frac{1}{2} M R^2 = \frac{1}{2} \times 4 \times (0.2)^2 = 0.08 \, \text{kg m}^2 \).

\( L = I \omega = 0.08 \times 25 = 2 \, \text{kg m}^2/\text{s} \).

1.5 kg m²/s
2.0 kg m²/s
2.5 kg m²/s
3.0 kg m²/s
2

A \( 4 \, \text{kg} \) block moves with a velocity of \( -2 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \, \text{m/s} \). 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 = \( \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6 \, \text{m/s} \).

3.2 m/s
3.6 m/s
4.0 m/s
4.5 m/s
2

Which of the following best defines a rigid body?

A rigid body is one where the distances between all pairs of particles remain constant, even under the influence of forces. This is an idealization, as real bodies deform slightly, but it’s a key assumption in rotational dynamics.

A body that can only undergo translational motion
A body with a fixed shape where distances between particles do not change
A body that deforms under external forces
A body with no mass distribution
2

A solid sphere of mass \( 5 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) has an angular momentum of \( 8 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

\( I = \frac{2}{5} M R^2 = \frac{2}{5} \times 5 \times (0.2)^2 = 0.08 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{8}{0.08} = 100 \, \text{rad/s} \).

80 rad/s
90 rad/s
100 rad/s
110 rad/s
3

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

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 3 & 0 & 0 \\ -5 & 2 & 0 \end{vmatrix} = \hat{\mathbf{k}} (3 \times 2 - 0 \times (-5)) = 6 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

Z-component = \( 6 \, \text{kg m}^2/\text{s} \).

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

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