NEET System of Particles and Rotational Motion — Set 12

Question 1

A solid cylinder of mass $ 3 \, \text{kg} $ and radius $ 0.4 \, \text{m} $ has an angular momentum of $ 12 \, \text{kg m}^2/\text{s} $. What is its angular velocity?

Accepted Answer

$ I = \frac{1}{2} M R^2 = \frac{1}{2} \times 3 \times (0.4)^2 = 0.24 \, \text{kg m}^2 $. $ \omega = \frac{L}{I} = \frac{12}{0.24} = 50 \, \text{rad/s} $.

Question 2

Three particles of masses $ 2 \, \text{kg} $, $ 3 \, \text{kg} $, and $ 5 \, \text{kg} $ are placed at coordinates $ (0, 0) $, $ (4, 0) $, and $ (2, 3) $ respectively. What is the x-coordinate of their center of mass?

Accepted Answer

Using the formula for the x-coordinate of the center of mass: $ X = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} $. Masses: $ m_1 = 2 \, \text{kg} $, $ m_2 = 3 \, \text{kg} $, $ m_3 = 5 \, \text{kg} $. Coordinates: $ x_1 = 0 $, $ x_2 = 4 $, $ x_3 = 2 $. $ X = \frac{(2 \times 0) + (3 \times 4) + (5 \times 2)}{2 + 3 + 5} = \frac{0 + 12 + 10}{10} = \frac{22}{10} = 2.2 \, \text{m} $.

Question 3

Why is torque zero when the force is applied along the line passing through the pivot point?

Accepted Answer

Torque $ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} $ depends on the perpendicular distance ($ r \sin \theta $). If the force acts along the line through the pivot, $ \theta = 0^\circ $ or $ 180^\circ $, so $ \sin \theta = 0 $, making torque zero.

NEET Physics: System of Particles and Rotational Motion — Practice Set 12

Q1. A solid cylinder of mass \( 3 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has an angular momentum of \( 12 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q2. Three particles of masses \( 2 \, \text{kg} \), \( 3 \, \text{kg} \), and \( 5 \, \text{kg} \) are placed at coordinates \( (0, 0) \), \( (4, 0) \), and \( (2, 3) \) respectively. What is the x-coordinate of their center of mass?

Q3. Why is torque zero when the force is applied along the line passing through the pivot point?

Q4. A torque of \( 18 \, \text{Nm} \) acts on a disk with moment of inertia \( 6 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 4 \, \text{s} \)?

Q5. A force \( \mathbf{F} = 4 \, \hat{\mathbf{i}} - 6 \, \hat{\mathbf{j}} \, \text{N} \) acts at \( \mathbf{r} = 5 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of the torque about the origin?

Q6. A cylinder of mass \( 10 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) rolls without slipping. If its angular momentum about its axis is \( 25 \, \text{kg m}^2/\text{s} \), what is its angular velocity?

Q7. Two particles of masses \( 7 \, \text{kg} \) and \( 3 \, \text{kg} \) are at \( (0, 6) \) and \( (5, 0) \) respectively. What is the distance of their center of mass from the origin?

Q8. A \( 2 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 5 \, \hat{\mathbf{i}} - 2 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 3 \, \hat{\mathbf{j}} \, \text{m} \). What is the z-component of its angular momentum?

Q9. A solid sphere of mass \( 4 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) has an angular momentum of \( 7.2 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q10. Vectors \( \mathbf{a} = 3 \, \hat{\mathbf{i}} - 5 \, \hat{\mathbf{j}} \) and \( \mathbf{b} = 2 \, \hat{\mathbf{i}} + 4 \, \hat{\mathbf{j}} \) are given. What is the magnitude of \( \mathbf{a} \times \mathbf{b} \)?