NEET System of Particles and Rotational Motion — Set 13

Question 1

A thin ring of mass $ 1 \, \text{kg} $ and radius $ 0.2 \, \text{m} $ rotates about its center. If its kinetic energy is $ 4 \, \text{J} $, what is its angular speed?

Accepted Answer

KE: $ K = \frac{1}{2} I \omega^2 $. For a ring: $ I = M R^2 = 1 \times (0.2)^2 = 0.04 \, \text{kg m}^2 $. $ 4 = \frac{1}{2} \times 0.04 \times \omega^2 \Rightarrow 8 = 0.04 \omega^2 \Rightarrow \omega^2 = 200 \Rightarrow \omega = \sqrt{200} \approx 14.14 \, \text{rad/s} $.

Question 2

A thin ring of mass $ 3 \, \text{kg} $ and radius $ 0.5 \, \text{m} $ rotates at $ 8 \, \text{rad/s} $. What is its angular momentum about its axis?

Accepted Answer

$ I = M R^2 = 3 \times (0.5)^2 = 0.75 \, \text{kg m}^2 $. $ L = I \omega = 0.75 \times 8 = 6 \, \text{kg m}^2/\text{s} $.

Question 3

A uniform triangular lamina has vertices at $ (0, 0) $, $ (6, 0) $, and $ (0, 8) $. What are the coordinates of its center of mass?

Accepted Answer

For a uniform triangular lamina, the center of mass is at the centroid, given by the average of the vertices. $ X = \frac{0 + 6 + 0}{3} = 2 $, $ Y = \frac{0 + 0 + 8}{3} = \frac{8}{3} \approx 2.67 \, \text{m} $. Coordinates: $ (2, \frac{8}{3}) $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 13

Q1. A thin ring of mass \( 1 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates about its center. If its kinetic energy is \( 4 \, \text{J} \), what is its angular speed?

Q2. A thin ring of mass \( 3 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) rotates at \( 8 \, \text{rad/s} \). What is its angular momentum about its axis?

Q3. A uniform triangular lamina has vertices at \( (0, 0) \), \( (6, 0) \), and \( (0, 8) \). What are the coordinates of its center of mass?

Q4. A hollow cylinder of mass \( 5 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates about its axis at \( 10 \, \text{rad/s} \). What is its rotational kinetic energy?

Q5. A solid sphere of mass \( 3 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates at \( 15 \, \text{rad/s} \). What is its angular momentum about its axis?

Q6. In rotational dynamics, what role does torque play?

Q7. What is the effect of a force applied at the axis of rotation on a rigid body?

Q8. Vectors \( \mathbf{a} = 7 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = -3 \, \hat{\mathbf{i}} + 4 \, \hat{\mathbf{j}} \) are given. What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?

Q9. A \( 4 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 5 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = -3 \, \hat{\mathbf{i}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?

Q10. A wheel with moment of inertia \( 4 \, \text{kg m}^2 \) rotates at \( 5 \, \text{rad/s} \). A torque of \( 8 \, \text{Nm} \) acts for \( 3 \, \text{s} \). What is its final angular velocity?