NEET System of Particles and Rotational Motion — Set 15

Question 1

A torque of $ 16 \, \text{Nm} $ is applied to a wheel with moment of inertia $ 8 \, \text{kg m}^2 $. What is its angular acceleration?

Accepted Answer

$ \tau = I \alpha $. $ \tau = 16 \, \text{Nm} $, $ I = 8 \, \text{kg m}^2 $. $ \alpha = \frac{\tau}{I} = \frac{16}{8} = 2 \, \text{rad/s}^2 $.

Question 2

A solid cylinder of mass $ 5 \, \text{kg} $ and radius $ 0.4 \, \text{m} $ rolls without slipping with a linear speed of $ 2.4 \, \text{m/s} $. What is its angular velocity?

Accepted Answer

$ v = \omega r $. $ v = 2.4 \, \text{m/s} $, $ r = 0.4 \, \text{m} $. $ \omega = \frac{v}{r} = \frac{2.4}{0.4} = 6 \, \text{rad/s} $.

Question 3

Three particles of masses $ 5 \, \text{kg} $, $ 10 \, \text{kg} $, and $ 15 \, \text{kg} $ are at $ (0, 0) $, $ (2, 4) $, and $ (6, 2) $ respectively. What is the x-coordinate of their center of mass?

Accepted Answer

Formula: $ X = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} $. Masses: $ 5, 10, 15 \, \text{kg} $; x-coordinates: $ 0, 2, 6 $. $ X = \frac{(5 \times 0) + (10 \times 2) + (15 \times 6)}{5 + 10 + 15} = \frac{0 + 20 + 90}{30} = \frac{110}{30} \approx 3.67 \, \text{m} $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 15

Q1. A torque of \( 16 \, \text{Nm} \) is applied to a wheel with moment of inertia \( 8 \, \text{kg m}^2 \). What is its angular acceleration?

Q2. A solid cylinder of mass \( 5 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) rolls without slipping with a linear speed of \( 2.4 \, \text{m/s} \). What is its angular velocity?

Q3. Three particles of masses \( 5 \, \text{kg} \), \( 10 \, \text{kg} \), and \( 15 \, \text{kg} \) are at \( (0, 0) \), \( (2, 4) \), and \( (6, 2) \) respectively. What is the x-coordinate of their center of mass?

Q4. Three particles of masses \( 2 \, \text{kg} \), \( 4 \, \text{kg} \), and \( 6 \, \text{kg} \) are at \( (0, 2) \), \( (3, 0) \), and \( (1, 4) \) respectively. What is the x-coordinate of their center of mass?

Q5. A solid sphere of mass \( 1 \, \text{kg} \) and radius \( 0.8 \, \text{m} \) has a kinetic energy of \( 12.8 \, \text{J} \). What is its angular speed?

Q6. Three particles of masses \( 1 \, \text{kg} \), \( 2 \, \text{kg} \), and \( 3 \, \text{kg} \) are located at \( (1, 1) \), \( (2, 3) \), and \( (4, 2) \) respectively. What is the y-coordinate of their center of mass?

Q7. A force \( \mathbf{F} = -2 \, \hat{\mathbf{i}} + 5 \, \hat{\mathbf{j}} \, \text{N} \) acts at \( \mathbf{r} = 4 \, \hat{\mathbf{i}} - 3 \, \hat{\mathbf{j}} \, \text{m} \). What is the magnitude of the torque about the origin?

Q8. Two particles of masses \( 8 \, \text{kg} \) and \( 2 \, \text{kg} \) are at \( (0, 5) \) and \( (10, 0) \) respectively. What is the distance of their center of mass from the origin?

Q9. A uniform circular disk of radius \( 3 \, \text{m} \) and mass \( 12 \, \text{kg} \) has its center at \( (0, 0) \). What is the position of its center of mass?

Q10. What determines the magnitude of torque on a rigid body?