NEET System of Particles and Rotational Motion — Set 19

Question 1

A solid cylinder of mass $ 2 \, \text{kg} $ and radius $ 0.5 \, \text{m} $ has a kinetic energy of $ 10 \, \text{J} $. What is its angular speed?

Accepted Answer

$ I = \frac{1}{2} M R^2 = \frac{1}{2} \times 2 \times (0.5)^2 = 0.25 \, \text{kg m}^2 $. $ K = \frac{1}{2} I \omega^2 \Rightarrow 10 = \frac{1}{2} \times 0.25 \times \omega^2 \Rightarrow 20 = 0.25 \omega^2 \Rightarrow \omega^2 = 80 \Rightarrow \omega = \sqrt{80} \approx 8.94 \, \text{rad/s} $.

Question 2

A solid sphere of mass $ 5 \, \text{kg} $ and radius $ 0.2 \, \text{m} $ has an angular momentum of $ 8 \, \text{kg m}^2/\text{s} $. What is its angular velocity?

Accepted Answer

$ I = \frac{2}{5} M R^2 = \frac{2}{5} \times 5 \times (0.2)^2 = 0.08 \, \text{kg m}^2 $. $ \omega = \frac{L}{I} = \frac{8}{0.08} = 100 \, \text{rad/s} $.

Question 3

Two particles of masses $ 3 \, \text{kg} $ and $ 7 \, \text{kg} $ are at $ (1, 4) $ and $ (9, 2) $ respectively. What is the distance of their center of mass from the origin?

Accepted Answer

$ X = \frac{(3 \times 1) + (7 \times 9)}{3 + 7} = \frac{3 + 63}{10} = 6.6 $. $ Y = \frac{(3 \times 4) + (7 \times 2)}{3 + 7} = \frac{12 + 14}{10} = 2.6 $. Distance = $ \sqrt{(6.6)^2 + (2.6)^2} = \sqrt{43.56 + 6.76} = \sqrt{50.32} \approx 7.09 \, \text{m} $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 19

Q1. A solid cylinder of mass \( 2 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) has a kinetic energy of \( 10 \, \text{J} \). What is its angular speed?

Q2. A solid sphere of mass \( 5 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) has an angular momentum of \( 8 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q3. Two particles of masses \( 3 \, \text{kg} \) and \( 7 \, \text{kg} \) are at \( (1, 4) \) and \( (9, 2) \) respectively. What is the distance of their center of mass from the origin?

Q4. Which of the following best defines a rigid body?

Q5. A \( 4 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = -5 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 3 \, \hat{\mathbf{i}} \, \text{m} \). What is the z-component of its angular momentum?

Q6. A uniform rectangular plate of dimensions \( 3 \, \text{m} \times 5 \, \text{m} \) and mass \( 12 \, \text{kg} \) has one corner at \( (1, 1) \) along the x- and y-axes. What is the position of its center of mass?

Q7. A hollow cylinder of mass \( 4 \, \text{kg} \) and radius \( 0.6 \, \text{m} \) rotates at \( 5 \, \text{rad/s} \). What is its angular momentum about its axis?

Q8. A \( 4 \, \text{kg} \) block moves with a velocity of \( -2 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \, \text{m/s} \). What is the magnitude of the velocity of its center of mass?

Q9. A solid cylinder of mass \( 4 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates at \( 25 \, \text{rad/s} \). What is its angular momentum about its axis?

Q10. A solid sphere of mass \( 5 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rotates about an axis through its center. If its angular momentum is \( 9 \, \text{kg m}^2/\text{s} \), what is its angular velocity?