NEET System of Particles and Rotational Motion — Set 20

Question 1

A $ 3 \, \text{kg} $ particle moves with velocity $ \mathbf{v} = 6 \, \hat{\mathbf{i}} \, \text{m/s} $ at $ \mathbf{r} = 2 \, \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} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 0 & 2 & 0 \ 6 & 0 & 0 \end{vmatrix} = \hat{\mathbf{k}} (0 \times 0 - 2 \times 6) = -12 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} $. Magnitude = $ 12 \, \text{kg m}^2/\text{s} $.

Question 2

Vectors $ \mathbf{a} = 8 \, \hat{\mathbf{i}} - 3 \, \hat{\mathbf{j}} $ and $ \mathbf{b} = 2 \, \hat{\mathbf{i}} + 7 \, \hat{\mathbf{j}} $ are given. What is the magnitude of $ \mathbf{a} \times \mathbf{b} $?

Accepted Answer

$ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 8 & -3 & 0 \ 2 & 7 & 0 \end{vmatrix} = \hat{\mathbf{k}} (8 \times 7 - (-3) \times 2) = \hat{\mathbf{k}} (56 + 6) = 62 \, \hat{\mathbf{k}} $. Magnitude = $ 62 $.

Question 3

A wheel with moment of inertia $ 6 \, \text{kg m}^2 $ rotates at $ 3 \, \text{rad/s} $. A torque of $ 12 \, \text{Nm} $ acts for $ 3 \, \text{s} $. What is its final angular velocity?

Accepted Answer

$ \alpha = \frac{\tau}{I} = \frac{12}{6} = 2 \, \text{rad/s}^2 $. $ \Delta \omega = \alpha t = 2 \times 3 = 6 \, \text{rad/s} $. $ \omega = \omega_0 + \Delta \omega = 3 + 6 = 9 \, \text{rad/s} $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 20

Q1. A \( 3 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 6 \, \hat{\mathbf{i}} \, \text{m/s} \) at \( \mathbf{r} = 2 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?

Q2. Vectors \( \mathbf{a} = 8 \, \hat{\mathbf{i}} - 3 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = 2 \, \hat{\mathbf{i}} + 7 \, \hat{\mathbf{j}} \) are given. What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?

Q3. A wheel with moment of inertia \( 6 \, \text{kg m}^2 \) rotates at \( 3 \, \text{rad/s} \). A torque of \( 12 \, \text{Nm} \) acts for \( 3 \, \text{s} \). What is its final angular velocity?

Q4. What is the primary source of angular acceleration in a rigid body?

Q5. A \( 7 \, \text{kg} \) object moves with a velocity of \( 3 \, \hat{\mathbf{i}} - 4 \, \hat{\mathbf{j}} \, \text{m/s} \). What is the magnitude of the velocity of its center of mass?

Q6. A uniform rod of mass \( 2 \, \text{kg} \) and length \( 1 \, \text{m} \) is pivoted at one end. What is its moment of inertia about the pivot?

Q7. A hollow cylinder of mass \( 1 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) has a kinetic energy of \( 2 \, \text{J} \). What is its angular speed?

Q8. Vectors \( \mathbf{a} = -4 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = 2 \, \hat{\mathbf{i}} + 5 \, \hat{\mathbf{j}} \) are given. What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?

Q9. What characterizes pure translational motion of a rigid body?

Q10. A uniform triangular lamina has vertices at \( (0, 0) \), \( (5, 0) \), and \( (0, 6) \). What is the x-coordinate of its center of mass?