NEET System of Particles and Rotational Motion — Set 6

Question 1

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

Accepted Answer

$ \mathbf{L} = \mathbf{r} \times \mathbf{p} = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \ 0 & 4 & 0 \ 2 & -3 & 0 \end{vmatrix} = \hat{\mathbf{k}} (0 \times (-3) - 4 \times 2) = -8 \, \hat{\mathbf{k}} \, \text{kg m}^2/\text{s} $. Z-component = $ -8 \, \text{kg m}^2/\text{s} $.

Question 2

A particle rotates about a fixed axis at a radius of $ 0.9 \, \text{m} $ with a linear speed of $ 4.5 \, \text{m/s} $. What is its angular velocity?

Accepted Answer

$ v = \omega r $. $ v = 4.5 \, \text{m/s} $, $ r = 0.9 \, \text{m} $. $ \omega = \frac{v}{r} = \frac{4.5}{0.9} = 5 \, \text{rad/s} $.

Question 3

A torque of $ 24 \, \text{Nm} $ acts on a wheel with moment of inertia $ 6 \, \text{kg m}^2 $ starting from rest. What is its angular speed after $ 3 \, \text{s} $?

Accepted Answer

$ \alpha = \frac{\tau}{I} = \frac{24}{6} = 4 \, \text{rad/s}^2 $. $ \omega = \omega_0 + \alpha t = 0 + 4 \times 3 = 12 \, \text{rad/s} $.

NEET Physics: System of Particles and Rotational Motion — Practice Set 6

Q1. A \( 5 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = 2 \, \hat{\mathbf{i}} - 3 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 4 \, \hat{\mathbf{j}} \, \text{m} \). What is the z-component of its angular momentum?

Q2. A particle rotates about a fixed axis at a radius of \( 0.9 \, \text{m} \) with a linear speed of \( 4.5 \, \text{m/s} \). What is its angular velocity?

Q3. A torque of \( 24 \, \text{Nm} \) acts on a wheel with moment of inertia \( 6 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 3 \, \text{s} \)?

Q4. A uniform rod of mass \( 5 \, \text{kg} \) and length \( 1.2 \, \text{m} \) is pivoted at its center. What is its moment of inertia about the pivot?

Q5. A particle in a rigid body rotates at a distance of \( 0.7 \, \text{m} \) from a fixed axis with an angular velocity of \( 6 \, \text{rad/s} \). What is its linear speed?

Q6. A force \( \mathbf{F} = 5 \, \hat{\mathbf{i}} + 2 \, \hat{\mathbf{j}} \, \text{N} \) acts at \( \mathbf{r} = -1 \, \hat{\mathbf{i}} + 3 \, \hat{\mathbf{j}} \, \text{m} \). What is the torque about the origin?

Q7. A torque of \( 20 \, \text{Nm} \) acts on a wheel with moment of inertia \( 8 \, \text{kg m}^2 \) starting from rest. What is its angular speed after \( 5 \, \text{s} \)?

Q8. Why does a dancer spin faster when pulling their arms closer to their body?

Q9. A torque of \( 12 \, \text{Nm} \) acts on a wheel with moment of inertia \( 3 \, \text{kg m}^2 \) for \( 4 \, \text{s} \). If it starts from rest, what is its final angular speed?

Q10. A particle moves in a circle of radius \( 0.5 \, \text{m} \) with an angular velocity of \( 4 \, \text{rad/s} \). What is the magnitude of its linear velocity?

Q11. A solid sphere of mass \( 2 \, \text{kg} \) and radius \( 0.3 \, \text{m} \) rolls without slipping with an angular velocity of \( 10 \, \text{rad/s} \). What is its linear speed?

Q12. What is the effect of increasing the mass of a rotating body on its angular momentum, if angular velocity remains constant?

Q13. A uniform rod of mass \( 3 \, \text{kg} \) and length \( 1.5 \, \text{m} \) is pivoted at one end. What is its moment of inertia about the pivot?

Q14. A particle rotates about a fixed axis at a radius of \( 0.6 \, \text{m} \) with a linear speed of \( 7.2 \, \text{m/s} \). What is its angular velocity?

Q15. A \( 3 \, \text{kg} \) particle moves with velocity \( \mathbf{v} = -4 \, \hat{\mathbf{j}} \, \text{m/s} \) at \( \mathbf{r} = 6 \, \hat{\mathbf{i}} \, \text{m} \). What is the magnitude of its angular momentum about the origin?