NEET System of Particles and Rotational Motion — Set 22

Question 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?

Accepted Answer

$ \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} $.

Question 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?

Accepted Answer

$ \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} $.

Question 3

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

Accepted Answer

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.

NEET Physics: System of Particles and Rotational Motion — Practice Set 22

Q1. 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?

Q2. 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?

Q3. What is the effect of doubling the radius of a rotating disk on its moment of inertia, assuming mass remains constant?

Q4. In a rigid body undergoing both translation and rotation, what describes the motion of any point?

Q5. 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?

Q6. 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 \( \mathbf{a} \times \mathbf{b} \)?

Q7. 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?

Q8. 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?

Q9. 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?

Q10. 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?