System of Particles and Rotational Motion Chapter-Wise Test 11

Correct answer Carries: 4.

Wrong Answer Carries: -1.

Vectors \( \mathbf{a} = 4 \, \hat{\mathbf{i}} - 2 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = 3 \, \hat{\mathbf{i}} + 5 \, \hat{\mathbf{j}} \) are given. What is the direction of \( \mathbf{a} \times \mathbf{b} \)?

\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 4 & -2 & 0 \\ 3 & 5 & 0 \end{vmatrix} = \hat{\mathbf{k}} (4 \times 5 - (-2) \times 3) = \hat{\mathbf{k}} (20 + 6) = 26 \, \hat{\mathbf{k}} \).

Direction is along the positive z-axis (\( \hat{\mathbf{k}} \)).

\( -\hat{\mathbf{k}} \)
\( \hat{\mathbf{i}} \)
\( \hat{\mathbf{j}} \)
\( \hat{\mathbf{k}} \)
4

A uniform circular disk of radius \( 4 \, \text{m} \) and mass \( 8 \, \text{kg} \) has its center at \( (-1, 3) \). What is the position of its center of mass?

For a uniform circular disk, the center of mass is at its geometric center.

Given center at \( (-1, 3) \), CM position = \( (-1, 3) \).

\( (0, 2) \)
\( (-1, 2) \)
\( (-1, 3) \)
\( (0, 3) \)
3

A solid sphere of mass \( 5 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates at \( 10 \, \text{rad/s} \). What is its angular momentum about its axis?

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

\( L = I \omega = 0.08 \times 10 = 0.8 \, \text{kg m}^2/\text{s} \).

0.6 kg m²/s
0.7 kg m²/s
0.8 kg m²/s
0.9 kg m²/s
3

Two particles of masses \( 4 \, \text{kg} \) and \( 6 \, \text{kg} \) are at \( (0, 7) \) and \( (8, 3) \) respectively. What is the distance of their center of mass from the origin?

\( X = \frac{(4 \times 0) + (6 \times 8)}{4 + 6} = \frac{48}{10} = 4.8 \).

\( Y = \frac{(4 \times 7) + (6 \times 3)}{4 + 6} = \frac{28 + 18}{10} = 4.6 \).

Distance = \( \sqrt{(4.8)^2 + (4.6)^2} = \sqrt{23.04 + 21.16} = \sqrt{44.2} \approx 6.65 \, \text{m} \).

6.4 m
6.65 m
6.8 m
7.0 m
2

Two particles of masses \( 5 \, \text{kg} \) and \( 3 \, \text{kg} \) are at \( (0, 4) \) and \( (8, 0) \) respectively. What is the distance of their center of mass from the origin?

\( X = \frac{(5 \times 0) + (3 \times 8)}{5 + 3} = \frac{24}{8} = 3 \).

\( Y = \frac{(5 \times 4) + (3 \times 0)}{5 + 3} = \frac{20}{8} = 2.5 \).

Distance = \( \sqrt{X^2 + Y^2} = \sqrt{3^2 + (2.5)^2} = \sqrt{9 + 6.25} = \sqrt{15.25} \approx 3.9 \, \text{m} \).

3.5 m
3.9 m
4.2 m
4.5 m
2

A force \( \mathbf{F} = 6 \, \hat{\mathbf{i}} - 2 \, \hat{\mathbf{j}} \, \text{N} \) acts at \( \mathbf{r} = 3 \, \hat{\mathbf{i}} + 4 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of the torque about the origin?

\( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 3 & 4 & 0 \\ 6 & -2 & 0 \end{vmatrix} = \hat{\mathbf{k}} (3 \times (-2) - 4 \times 6) = \hat{\mathbf{k}} (-6 - 24) = -30 \, \hat{\mathbf{k}} \, \text{Nm} \).

Magnitude = \( 30 \, \text{Nm} \).

28 Nm
29 Nm
30 Nm
31 Nm
3

A solid sphere of mass \( 2 \, \text{kg} \) and radius \( 0.1 \, \text{m} \) has an angular momentum of \( 4 \, \text{kg m}^2/\text{s} \) about its axis. What is its angular velocity?

\( L = I \omega \).

For a sphere: \( I = \frac{2}{5} M R^2 = \frac{2}{5} \times 2 \times (0.1)^2 = 0.008 \, \text{kg m}^2 \).

\( \omega = \frac{L}{I} = \frac{4}{0.008} = 500 \, \text{rad/s} \).

400 rad/s
450 rad/s
500 rad/s
550 rad/s
3

What is the direction of the angular velocity vector for a body rotating about a fixed axis?

The angular velocity vector is directed along the axis of rotation, following the right-hand rule (curl fingers in the direction of rotation, thumb points along the vector).

Perpendicular to the plane of rotation
Along the axis of rotation
Tangential to the circular path of particles
Radial towards the center of rotation
2

A uniform square plate of side \( 6 \, \text{m} \) and mass \( 9 \, \text{kg} \) has one corner at \( (2, 2) \) along the x- and y-axes. What is the position of its center of mass?

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

Vertices: \( (2, 2), (8, 2), (2, 8), (8, 8) \).

CM: \( X = 2 + \frac{6}{2} = 5 \), \( Y = 2 + \frac{6}{2} = 5 \). Position: \( (5, 5) \, \text{m} \).

\( (4, 4) \)
\( (5, 4) \)
\( (5, 5) \)
\( (6, 5) \)
3

A \( 1 \, \text{kg} \) mass rotates in a circle of radius \( 0.5 \, \text{m} \) with a speed of \( 4 \, \text{m/s} \). What is its angular momentum about the center?

\( L = m v r \).

\( m = 1 \, \text{kg} \), \( v = 4 \, \text{m/s} \), \( r = 0.5 \, \text{m} \).

\( L = 1 \times 4 \times 0.5 = 2 \, \text{kg m}^2/\text{s} \).

1 kg m²/s
2 kg m²/s
3 kg m²/s
4 kg m²/s
2

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