System Of Particles And Rotational Motion Chapter-Wise Test 1

Correct answer Carries: 4.

Wrong Answer Carries: -1.

A force \( \mathbf{F} = 7 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \, \text{N} \) acts at \( \mathbf{r} = -2 \, \hat{\mathbf{i}} + 1 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of the torque about the origin?

\( \mathbf{\tau} = \mathbf{r} \times \mathbf{F} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ -2 & 1 & 0 \\ 7 & 3 & 0 \end{vmatrix} = \hat{\mathbf{k}} ((-2) \times 3 - 1 \times 7) = \hat{\mathbf{k}} (-6 - 7) = -13 \, \hat{\mathbf{k}} \, \text{Nm} \).

Magnitude = \( 13 \, \text{Nm} \).

12 Nm
13 Nm
14 Nm
15 Nm
2

A wheel with moment of inertia \( 5 \, \text{kg m}^2 \) rotates at \( 3 \, \text{rad/s} \). A torque of \( 10 \, \text{Nm} \) acts for \( 5 \, \text{s} \). What is its final angular velocity?

\( \alpha = \frac{\tau}{I} = \frac{10}{5} = 2 \, \text{rad/s}^2 \).

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

\( \omega = \omega_0 + \Delta \omega = 3 + 10 = 13 \, \text{rad/s} \).

11 rad/s
12 rad/s
13 rad/s
14 rad/s
3

Two vectors are given as \( \mathbf{a} = 6 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = -4 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \). What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?

\( \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 6 & 3 & 0 \\ -4 & 2 & 0 \end{vmatrix} = \hat{\mathbf{k}} (6 \times 2 - 3 \times (-4)) = \hat{\mathbf{k}} (12 + 12) = 24 \, \hat{\mathbf{k}} \).

Magnitude = \( 24 \).

22
23
24
25
3

A solid sphere of mass \( 2 \, \text{kg} \) and radius \( 0.6 \, \text{m} \) rotates about its center at \( 5 \, \text{rad/s} \). What is its rotational kinetic energy?

\( I = \frac{2}{5} M R^2 = \frac{2}{5} \times 2 \times (0.6)^2 = 0.288 \, \text{kg m}^2 \).

\( K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.288 \times (5)^2 = 0.144 \times 25 = 3.6 \, \text{J} \).

3.2 J
3.6 J
4.0 J
4.4 J
2

A wheel with moment of inertia \( 8 \, \text{kg m}^2 \) rotates at \( 2 \, \text{rad/s} \). A torque of \( 16 \, \text{Nm} \) acts for \( 3 \, \text{s} \). What is its final angular velocity?

\( \alpha = \frac{\tau}{I} = \frac{16}{8} = 2 \, \text{rad/s}^2 \).

\( \Delta \omega = \alpha t = 2 \times 3 = 6 \, \text{rad/s} \).

\( \omega = \omega_0 + \Delta \omega = 2 + 6 = 8 \, \text{rad/s} \).

6 rad/s
7 rad/s
8 rad/s
9 rad/s
3

A uniform square lamina of side \( 4 \, \text{m} \) and mass \( 8 \, \text{kg} \) has one corner at the origin and lies 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: \( (0, 0), (4, 0), (0, 4), (4, 4) \).

CM: \( X = \frac{0 + 4}{2} = 2 \), \( Y = \frac{0 + 4}{2} = 2 \). Position: \( (2, 2) \, \text{m} \).

\( (1, 1) \)
\( (2, 2) \)
\( (3, 3) \)
\( (4, 4) \)
2

What is the effect of doubling the radius of a rotating disk on its moment of inertia, assuming mass remains constant?

For a disk, \( I = \frac{1}{2} M R^2 \). Doubling \( R \) increases \( R^2 \) by a factor of 4, so \( I \) increases by a factor of 4 if \( M \) is constant.

It doubles
It quadruples
It halves
It remains the same
2

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

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ -2 & 0 & 0 \\ 3 & 5 & 0 \end{vmatrix} = \hat{\mathbf{k}} ((-2) \times 5 - 0 \times 3) = -10 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

Z-component = \( -10 \, \text{kg m}^2/\text{s} \).

-12 kg m²/s
-10 kg m²/s
-8 kg m²/s
-6 kg m²/s
2

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

\( \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ -4 & 0 & 0 \\ 0 & 3 & 0 \end{vmatrix} = \hat{\mathbf{k}} ((-4) \times 3 - 0 \times 0) = -12 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} \).

Magnitude = \( 12 \, \text{kg m}^2/\text{s} \).

10 kg m²/s
11 kg m²/s
12 kg m²/s
13 kg m²/s
3

In a rigid body undergoing both translation and rotation, what describes the motion of any point?

The motion of any point is a combination of the translational velocity of the body (same for all points) and the rotational velocity about the axis, varying with distance from the axis.

Pure translational velocity only
Pure rotational velocity only
Combination of translational and rotational velocities
Zero velocity at all points
3

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