NEET System of Particles and Rotational Motion — Set 21

Question 1

A $ 3 \, \text{kg} $ particle has a velocity $ \mathbf{v} = 2 \, \hat{\mathbf{i}} \, \text{m/s} $ at position $ \mathbf{r} = 4 \, \hat{\mathbf{j}} \, \text{m} $. What is the magnitude of its angular momentum about the origin?

Accepted Answer

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

Question 2

In the expression $ \mathbf{v} = \mathbf{\omega} \times \mathbf{r} $, what does $ \mathbf{v} $ represent?

Accepted Answer

The equation $ \mathbf{v} = \mathbf{\omega} \times \mathbf{r} $ gives the linear velocity of a particle in rotational motion, where $ \mathbf{\omega} $ is the angular velocity and $ \mathbf{r} $ is the position vector from the axis.

Question 3

A uniform rod of mass $ 5 \, \text{kg} $ and length $ 2 \, \text{m} $ is pivoted at its center. What is its moment of inertia about the pivot?

Accepted Answer

For a rod pivoted at its center: $ I = \frac{1}{12} M L^2 $. $ M = 5 \, \text{kg} $, $ L = 2 \, \text{m} $. $ I = \frac{1}{12} \times 5 \times (2)^2 = \frac{20}{12} = \frac{5}{3} \approx 1.67 \, \text{kg m}^2 $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 21

Q1. A \( 3 \, \text{kg} \) particle has a velocity \( \mathbf{v} = 2 \, \hat{\mathbf{i}} \, \text{m/s} \) at position \( \mathbf{r} = 4 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?

Q2. In the expression \( \mathbf{v} = \mathbf{\omega} \times \mathbf{r} \), what does \( \mathbf{v} \) represent?

Q3. A uniform rod of mass \( 5 \, \text{kg} \) and length \( 2 \, \text{m} \) is pivoted at its center. What is its moment of inertia about the pivot?

Q4. A \( 3 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 6 \, \hat{\mathbf{i}} - 4 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 2 \, \hat{\mathbf{j}} \, \text{m} \). What is the z-component of its angular momentum?

Q5. What happens to the angular velocity of a rigid body if the net external torque is zero?

Q6. Which of the following is a requirement for a rigid body to be in mechanical equilibrium?

Q7. A thin ring of mass \( 3 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) rotates about its center at \( 5 \, \text{rad/s} \). What is its rotational kinetic energy?

Q8. A solid sphere of mass \( 6 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has an angular momentum of \( 11.52 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q9. A thin ring of mass \( 1 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has a kinetic energy of \( 8 \, \text{J} \). What is its angular speed?

Q10. A torque of \( 25 \, \text{Nm} \) acts on a wheel with moment of inertia \( 5 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 2 \, \text{s} \)?