NEET System of Particles and Rotational Motion — Set 8

Question 1

What is the relationship between torque and angular acceleration for a rigid body?

Accepted Answer

Torque is directly proportional to angular acceleration ($ \mathbf{\tau} = I \mathbf{\alpha} $), with moment of inertia as the constant of proportionality.

Question 2

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

Accepted Answer

$ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 2 & 3 & 0 \ -3 & 4 & 0 \end{vmatrix} = \hat{\mathbf{k}} (2 \times 4 - 3 \times (-3)) = \hat{\mathbf{k}} (8 + 9) = 17 \, \hat{\mathbf{k}} $. Magnitude = $ |\mathbf{a} \times \mathbf{b}| = 17 $.

Question 3

A uniform rod of mass $ 6 \, \text{kg} $ and length $ 3 \, \text{m} $ is pivoted at one end. What is its moment of inertia about the pivot?

Accepted Answer

For a rod pivoted at one end: $ I = \frac{1}{3} M L^2 $. $ M = 6 \, \text{kg} $, $ L = 3 \, \text{m} $. $ I = \frac{1}{3} \times 6 \times (3)^2 = 2 \times 9 = 18 \, \text{kg m}^2 $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 8

Q1. What is the relationship between torque and angular acceleration for a rigid body?

Q2. Two vectors are given as \( \mathbf{a} = 2 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = -3 \, \hat{\mathbf{i}} + 4 \, \hat{\mathbf{j}} \). What is the magnitude of their vector product \( \mathbf{a} \times \mathbf{b} \)?

Q3. A uniform rod of mass \( 6 \, \text{kg} \) and length \( 3 \, \text{m} \) is pivoted at one end. What is its moment of inertia about the pivot?

Q4. What does the moment of inertia of a rigid body depend on?

Q5. A hollow cylinder 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. Why does a spinning top precess when tilted?

Q7. A uniform disk of mass \( 3 \, \text{kg} \) and radius \( 0.6 \, \text{m} \) rotates about its center. What is its moment of inertia?

Q8. Which statement is true about the vector product of two vectors?

Q9. A hollow sphere of mass \( 2 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) has an angular momentum of \( 6 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q10. What is the key assumption about distances in a rigid body during rotational motion?