NEET System of Particles and Rotational Motion — Set 5

Question 1

Three particles of masses $ 4 \, \text{kg} $, $ 8 \, \text{kg} $, and $ 12 \, \text{kg} $ are at $ (1, 0) $, $ (3, 2) $, and $ (0, 4) $ 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: $ 4, 8, 12 \, \text{kg} $; y-coordinates: $ 0, 2, 4 $. $ Y = \frac{(4 \times 0) + (8 \times 2) + (12 \times 4)}{4 + 8 + 12} = \frac{0 + 16 + 48}{24} = \frac{64}{24} = 2.67 \, \text{m} $.

Question 2

A system of two particles, $ 3 \, \text{kg} $ at $ (1, 2) $ and $ 1 \, \text{kg} $ at $ (5, 6) $, moves as a rigid body. What is the distance of the center of mass from the origin?

Accepted Answer

$ X = \frac{(3 \times 1) + (1 \times 5)}{3 + 1} = \frac{3 + 5}{4} = 2 $. $ Y = \frac{(3 \times 2) + (1 \times 6)}{3 + 1} = \frac{6 + 6}{4} = 3 $. Distance = $ \sqrt{X^2 + Y^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6 \, \text{m} $.

Question 3

A solid cylinder of mass $ 3 \, \text{kg} $ and radius $ 0.1 \, \text{m} $ has a kinetic energy of $ 4.5 \, \text{J} $. What is its angular speed?

Accepted Answer

$ I = \frac{1}{2} M R^2 = \frac{1}{2} \times 3 \times (0.1)^2 = 0.015 \, \text{kg m}^2 $. $ K = \frac{1}{2} I \omega^2 \Rightarrow 4.5 = \frac{1}{2} \times 0.015 \times \omega^2 \Rightarrow 9 = 0.015 \omega^2 \Rightarrow \omega^2 = 600 \Rightarrow \omega = \sqrt{600} \approx 24.5 \, \text{rad/s} $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 5

Q1. Three particles of masses \( 4 \, \text{kg} \), \( 8 \, \text{kg} \), and \( 12 \, \text{kg} \) are at \( (1, 0) \), \( (3, 2) \), and \( (0, 4) \) respectively. What is the y-coordinate of their center of mass?

Q2. A system of two particles, \( 3 \, \text{kg} \) at \( (1, 2) \) and \( 1 \, \text{kg} \) at \( (5, 6) \), moves as a rigid body. What is the distance of the center of mass from the origin?

Q3. A solid cylinder of mass \( 3 \, \text{kg} \) and radius \( 0.1 \, \text{m} \) has a kinetic energy of \( 4.5 \, \text{J} \). What is its angular speed?

Q4. A \( 4 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 3 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 2 \, \hat{\mathbf{i}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?

Q5. A uniform circular ring of radius \( 2 \, \text{m} \) and mass \( 5 \, \text{kg} \) has its center at \( (3, 4) \). What is the position of its center of mass?

Q6. A solid cylinder of mass \( 4 \, \text{kg} \) and radius \( 0.5 \, \text{m} \) has a kinetic energy of \( 50 \, \text{J} \). What is its angular speed?

Q7. A thin ring of mass \( 6 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rotates at \( 5 \, \text{rad/s} \). What is its angular momentum?

Q8. A torque of \( 10 \, \text{Nm} \) is applied to a wheel with moment of inertia \( 5 \, \text{kg m}^2 \). What is its angular acceleration?

Q9. A thin ring of mass \( 4 \, \text{kg} \) and radius \( 0.6 \, \text{m} \) has an angular momentum of \( 12 \, \text{kg m}^2/\text{s} \). What is its angular velocity?

Q10. A \( 1 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 3 \, \hat{\mathbf{i}} + 4 \, \hat{\mathbf{j}} \, \text{m/s} \) at position \( \mathbf{r} = 2 \, \hat{\mathbf{j}} \, \text{m} \). What is the z-component of its angular momentum about the origin?

Q11. A \( 5 \, \text{kg} \) object moves with a velocity of \( 4 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \, \text{m/s} \). What is the magnitude of the velocity of its center of mass?

Q12. A \( 2 \, \text{kg} \) mass rotates in a circle of radius \( 0.4 \, \text{m} \) with a speed of \( 6 \, \text{m/s} \). What is its angular momentum about the center?

Q13. A uniform rod of mass \( 4 \, \text{kg} \) and length \( 2.5 \, \text{m} \) is pivoted at its center. What is its moment of inertia about the pivot?

Q14. A hollow cylinder of mass \( 4 \, \text{kg} \) and radius \( 0.6 \, \text{m} \) rolls without slipping with a linear speed of \( 3 \, \text{m/s} \). What is its angular velocity?

Q15. A torque of \( 12 \, \text{Nm} \) is applied to a disk with moment of inertia \( 4 \, \text{kg m}^2 \). What is its angular acceleration?