NEET System of Particles and Rotational Motion — Set 11

Question 1

Two particles of masses $ 4 \, \text{kg} $ and $ 6 \, \text{kg} $ are at $ (0, 7) $ and $ (8, 3) $ respectively. What is the distance of their center of mass from the origin?

Accepted Answer

$ X = \frac{(4 \times 0) + (6 \times 8)}{4 + 6} = \frac{48}{10} = 4.8 $. $ Y = \frac{(4 \times 7) + (6 \times 3)}{4 + 6} = \frac{28 + 18}{10} = 4.6 $. Distance = $ \sqrt{(4.8)^2 + (4.6)^2} = \sqrt{23.04 + 21.16} = \sqrt{44.2} \approx 6.65 \, \text{m} $.

Question 2

A solid sphere of mass $ 5 \, \text{kg} $ and radius $ 0.2 \, \text{m} $ rotates at $ 10 \, \text{rad/s} $. What is its angular momentum about its axis?

Accepted Answer

$ I = \frac{2}{5} M R^2 = \frac{2}{5} \times 5 \times (0.2)^2 = 0.08 \, \text{kg m}^2 $. $ L = I \omega = 0.08 \times 10 = 0.8 \, \text{kg m}^2/\text{s} $.

Question 3

A solid sphere of mass $ 2 \, \text{kg} $ and radius $ 0.1 \, \text{m} $ has an angular momentum of $ 4 \, \text{kg m}^2/\text{s} $ about its axis. What is its angular velocity?

Accepted Answer

$ L = I \omega $. For a sphere: $ I = \frac{2}{5} M R^2 = \frac{2}{5} \times 2 \times (0.1)^2 = 0.008 \, \text{kg m}^2 $. $ \omega = \frac{L}{I} = \frac{4}{0.008} = 500 \, \text{rad/s} $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 11

Q1. Two particles of masses \( 4 \, \text{kg} \) and \( 6 \, \text{kg} \) are at \( (0, 7) \) and \( (8, 3) \) respectively. What is the distance of their center of mass from the origin?

Q2. A solid sphere of mass \( 5 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates at \( 10 \, \text{rad/s} \). What is its angular momentum about its axis?

Q3. A solid sphere of mass \( 2 \, \text{kg} \) and radius \( 0.1 \, \text{m} \) has an angular momentum of \( 4 \, \text{kg m}^2/\text{s} \) about its axis. What is its angular velocity?

Q4. A uniform circular disk of radius \( 4 \, \text{m} \) and mass \( 8 \, \text{kg} \) has its center at \( (-1, 3) \). What is the position of its center of mass?

Q5. Vectors \( \mathbf{a} = 4 \, \hat{\mathbf{i}} - 2 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = 3 \, \hat{\mathbf{i}} + 5 \, \hat{\mathbf{j}} \) are given. What is the direction of \( \mathbf{a} \times \mathbf{b} \)?

Q6. A \( 1 \, \text{kg} \) mass rotates in a circle of radius \( 0.5 \, \text{m} \) with a speed of \( 4 \, \text{m/s} \). What is its angular momentum about the center?

Q7. A force \( \mathbf{F} = 6 \, \hat{\mathbf{i}} - 2 \, \hat{\mathbf{j}} \, \text{N} \) acts at \( \mathbf{r} = 3 \, \hat{\mathbf{i}} + 4 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of the torque about the origin?

Q8. A uniform square plate of side \( 6 \, \text{m} \) and mass \( 9 \, \text{kg} \) has one corner at \( (2, 2) \) along the x- and y-axes. What is the position of its center of mass?

Q9. What is the direction of the angular velocity vector for a body rotating about a fixed axis?

Q10. Two particles of masses \( 5 \, \text{kg} \) and \( 3 \, \text{kg} \) are at \( (0, 4) \) and \( (8, 0) \) respectively. What is the distance of their center of mass from the origin?