NEET System of Particles and Rotational Motion — Set 2

Question 1

A solid sphere of mass $ 1 \, \text{kg} $ and radius $ 0.2 \, \text{m} $ rotates about its center at $ 10 \, \text{rad/s} $. What is its rotational kinetic energy?

Accepted Answer

$ I = \frac{2}{5} M R^2 = \frac{2}{5} \times 1 \times (0.2)^2 = 0.016 \, \text{kg m}^2 $. $ K = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 0.016 \times (10)^2 = 0.008 \times 100 = 0.8 \, \text{J} $.

Question 2

Three particles of masses $ 2 \, \text{kg} $, $ 4 \, \text{kg} $, and $ 8 \, \text{kg} $ are at $ (0, 3) $, $ (4, 0) $, and $ (2, 5) $ respectively. What is the y-coordinate of their center of mass?

Accepted Answer

Formula: $ Y = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{m_1 + m_2 + m_3} $. Masses: $ 2, 4, 8 \, \text{kg} $; y-coordinates: $ 3, 0, 5 $. $ Y = \frac{(2 \times 3) + (4 \times 0) + (8 \times 5)}{2 + 4 + 8} = \frac{6 + 0 + 40}{14} = \frac{46}{14} \approx 3.29 \, \text{m} $.

Question 3

A thin uniform ring of mass $ 2 \, \text{kg} $ and radius $ 0.5 \, \text{m} $ rotates about its center. What is its moment of inertia?

Accepted Answer

For a thin ring: $ I = M R^2 $. $ M = 2 \, \text{kg} $, $ R = 0.5 \, \text{m} $. $ I = 2 \times (0.5)^2 = 0.5 \, \text{kg m}^2 $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 2

Q1. A solid sphere of mass \( 1 \, \text{kg} \) and radius \( 0.2 \, \text{m} \) rotates about its center at \( 10 \, \text{rad/s} \). What is its rotational kinetic energy?

Q2. Three particles of masses \( 2 \, \text{kg} \), \( 4 \, \text{kg} \), and \( 8 \, \text{kg} \) are at \( (0, 3) \), \( (4, 0) \), and \( (2, 5) \) respectively. What is the y-coordinate of their center of mass?

Q3. A thin uniform ring of mass \( 2 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) rotates about its center. What is its moment of inertia?

Q4. What happens to the angular momentum of a rigid body if its moment of inertia increases while no external torque acts?

Q5. A \( 4 \, \text{kg} \) mass rotates in a circle of radius \( 0.25 \, \text{m} \) with a linear speed of \( 2 \, \text{m/s} \). What is its angular momentum about the center?

Q6. A particle in a rigid body rotates at a distance of \( 0.8 \, \text{m} \) from a fixed axis with an angular velocity of \( 7 \, \text{rad/s} \). What is its linear speed?

Q7. A thin ring of mass \( 5 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has an angular momentum of \( 8 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q8. What condition indicates that a rigid body is not rotating?

Q9. A thin ring of mass \( 6 \, \text{kg} \) and radius \( 0.4 \, \text{m} \) has an angular momentum of \( 9.6 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q10. What is the significance of the perpendicular component of force in calculating torque?

Q11. A \( 6 \, \text{kg} \) object moves with a velocity of \( 2 \, \hat{\mathbf{i}} + 5 \, \hat{\mathbf{j}} \, \text{m/s} \). What is the magnitude of the velocity of its center of mass?

Q12. A wheel with moment of inertia \( 2 \, \text{kg m}^2 \) rotates at \( 4 \, \text{rad/s} \). A torque of \( 6 \, \text{Nm} \) acts for \( 4 \, \text{s} \). What is its final angular velocity?

Q13. A uniform rectangular lamina of dimensions \( 2 \, \text{m} \times 4 \, \text{m} \) and mass \( 10 \, \text{kg} \) has one corner at the origin along the x- and y-axes. What is the position of its center of mass?

Q14. A thin ring of mass \( 2 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) has a kinetic energy of \( 10 \, \text{J} \). What is its angular speed?

Q15. A \( 2 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 5 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 3 \, \hat{\mathbf{i}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?