## Motion Of System Of Particles And Rigid Body

## Rotation Of Rigid Bodies Multiple Choice Questions And Answers

**Question 1. A stone is tied with a massless rope and is rotated with uniform speed. The angular momentum of the stone is L. Keeping the angular velocity unchanged if the length of the rope is halved, its angular momentum will be**

- \(\frac{L}{4}\)
- \(\frac{L}{2}\)
- L
- 2L

**Answer:** 1. \(\frac{L}{4}\)

**Question 2. Vector representation of angular momentum (\(\vec{L}\)) is**

- \(\vec{L}=\vec{p} \times \vec{r}\)
- \(\vec{L}=\vec{r} \times \vec{p}\)
- \(\vec{L}=\vec{p}·\vec{r}\)
- \(\vec{L}=\vec{r}·\vec{p}\)

**Answer: **2. \(\vec{L}=\vec{r} \times \vec{p}\)

**Question 3. A thin circular ring of mass M and radius R is rotating about an axis perpendicular to the plane of the ring and passing through the centre, with an angular velocity ω. Two bodies each of mass m are placed gently on the ring. The angular velocity with which the ring is rotating now is given by,**

- \(\frac{\omega M}{M+m}\)
- \(\frac{2(M-2 m)}{(M+2 m)}\)
- \(\frac{\omega M}{M+2 m}\)
- \(\frac{\omega(M+2 m)}{M}\)

**Answer:** 3. \(\frac{\omega M}{M+2 m}\)

**Question 4. A particle of mass m is moving with a uniform velocity along a straight path parallel to the x-axis. The angular momentum of the particle with respect to the origin will be**

- Zero
- Constant
- Increased gradually
- Decreased gradually

**Answer:** 2. Constant

**Question 5. A disc of mass M and radius R is revolving with an angular velocity ω on a horizontal plane. What will be the magnitude of angular momentum of the disc about the origin O?**

- \(\frac{1}{2} M R^2 \omega\)
- \(M R^2 \omega\)
- \(\frac{3}{2} M R^2 \omega\)
- \(2 M R^2 \omega\)

**Answer:** 3. \(\frac{3}{2} M R^2 \omega\)

**Question 6. The angular velocity of a smooth sphere A moving on a frictionless horizontal surface is ω and the velocity of its centre of mass is v. When it undergoes elastic head-on collision with another identical sphere B at rest, then the angular velocities of the two spheres become ω _{A} and ω_{B} respectively. If friction is neglected, the relation between ω_{A} and ω_{B} will be**

- ω
_{A}< ω_{B} - ω
_{A}= ω_{B} - ω
_{A}= ω - ω
_{B}= ω

**Answer:** 3. ω_{A} = ω

**Question 7. The angular momentum of a moving body remains constant if**

- An external force acts on the body
- Pressure acts on the body
- An external torque acts on the body
- No external torque acts on the body

**Answer:** 4. No external torque acts on the body

**Question 8. Angular momentum is**

- Moment of momentum
- Product of mass and angular velocity
- Product of moment of inertia and velocity
- Moment in angular motion

**Answer:** 1. Moment of momentum

**Question 9. A particle performs uniform circular motion with an angular momentum L. If the frequency of the particle motion is doubled, the angular momentum becomes**

- 2L
- 4L
- \(\frac{L}{2}\)
- \(\frac{L}{4}\)

**Answer:** 1. 2L

**Question 10. If r denotes the distance between the sun and the earth, then the angular momentum of the earth around the sun is proportional to**

- \(\frac{1}{r}\)
- r
- √r
- r²

**Answer:** 4. r²

**Question 11. The dimensional formula of torque is**

- ML²T
^{-2} - M²LT
^{-1} - MLT
^{-2} - ML²T
^{-1}

**Answer:** 1. ML²T^{-2}

**Question 12. The torque of a force \(\vec{F}=-6 \hat{i}\) acting at a point \(\vec{r}=4 \hat{j}\) about origin will be**

- \(-24 \hat{k}\)
- \(24 \hat{k}\)
- \(24 \hat{j}\)
- \(24 \hat{i}\)

**Answer:** 1. \(-24 \hat{k}\)

**Question 13. The moment of inertia of a circular ring of mass m and radius r about the normal axis passing through its centre is**

- \(\frac{m r^2}{4}\)
- \(m r^2\)
- \(\frac{m r^2}{2}\)
- \(\frac{3}{4} m r^2\)

**Answer:** 2. \(m r^2\)

**Question 14. The moment of inertia of a circular wire of mass m and radius r about its diameter is**

- \(\frac{1}{2}\)mr²
- \(\frac{1}{4}\)mr²
- mr²
- 2mr²

**Answer:** 1. \(\frac{1}{2}\)mr²

**Question 15. Thicknesses of two iron discs of radii r and 4r are t and \(\frac{t}{4}\) respectively. If their moments of inertia are I _{1} and I_{2} respectively, then**

- \(I_2=64 I_1\)
- \(I_2=32 I_1\)
- \(I_2=16 I_1\)
- \(I_2=I_1\)

**Answer:** 1. \(I_2=64 I_1\)

**Question 16. The moment of inertia of a hollow cylinder of mass M and radius r about its own axis is**

- \(\frac{2}{3}\)Mr²
- \(\frac{2}{5}\)Mr²
- Mr²
- \(\frac{1}{2}\)Mr²

**Answer:** 3. Mr²

**Question 17. The moment of inertia of a disc is 100 g · cm². The disc rotates with an angular velocity 2 rad/s. The rotational; kinetic energy of the disc is**

- 100 erg
- 200 erg
- 400 erg
- 50 erg

**Answer:** 2. 200 erg

**Question 18. The moment of inertia of a circular disc of mass m and radius r about a perpendicular axis passing through its centre is**

- mr²
- \(\frac{m r^2}{4}\)
- \(\frac{m r^2}{2}\)
- \(\frac{5}{4}\) mr²

**Answer:** 3. \(\frac{m r^2}{2}\)

**Question 19. Radius of gyration of a ring of radius R about an axis passing through its centre and perpendicular to its plane is**

- \(\frac{5}{\sqrt 2}\) R
- \(\frac{R}{2}\)
- R
- \(\frac{R}{\sqrt 2}\)

**Answer:** 3. R

**Question 20. Radius of gyration of a disc of mass 50 g and radius 0.5 cm about an axis passing through its centre of gravity and perpendicular to the plane is**

- 6.54 cm
- 3.64 cm
- 0.35 cm
- 0.88 cm

**Answer:** 3. 0.35 cm

**Question 21. The moment of inertia of a disc is 100 g · cm². If the disc rotates with an angular velocity of 2 rad · s ^{-1}, the rotational kinetic energy of the disc is**

- 100 erg
- 200 erg
- 400 erg
- 50 erg

**Answer:** 2. 200 erg

**Question 22. A man stands on a rotating table stretching his arms. He is rotating with a definite angular velocity. Now, the man draws his arms closer. His moment of inertia is reduced to 75% of its initial value. The angular kinetic energy of the man**

- Will increase by 33.3%
- Will decrease by 33.3%
- Will increase by 25%
- Will decrease by 25%

**Answer:** 1. Will increase by 33.3%

**Question 23. The total KE of the sphere of mass M rolling with velocity v is**

- \(\frac{7}{10}\)mv²
- \(\frac{5}{6}\)mv²
- \(\frac{7}{5}\)mv²
- \(\frac{10}{7}\)mv²

**Answer:** 1. \(\frac{7}{10}\)mv²

**Question 24. A body of mass 10 kg moves with a velocity of 2 m/s along a circular path of radius 8 m. The power produced by the body will be**

- 10J/s
- 98 J/s
- 49 J/s
- Zero

**Answer:** 1. 10J/s

**Question 25. If a sphere is rolling, then the ratio of its rotational kinetic energy to the total kinetic energy is**

- 1:2
- 2:5
- 2:7
- 5:7

**Answer:** 3. 2:7

**Question 26. If no torque acts on a rotating body and if its moment of inertia decreases, the angular velocity ω of the body increases in such a manner that**

- \(\frac{1}{2}\)Iω² remains constant
- Iω remains constant
- \(\frac{1}{\omega}\) remains constant
- Iω² remains constant

**Answer:** 2. Iω remains constant

**Question 27. The angular momentum of a particle revolving with uniform speed is L. If the frequency of the particle is doubled and its kinetic energy is halved, then its angular momentum becomes**

- 2.5 L
- 0.25 L
- 5.0 L
- 0.50 L

**Answer:** 2. 0.25 L

**Question 28. A particle is revolving along a circular path with decreasing speed. Which one of the following is true for the particle?**

- The angular momentum of the particle is constant
- Only the direction of angular momentum of the particle is fixed
- Acceleration of the particle is always towards the centre
- The particle travels along a spiral path

**Answer:** 2. Only the direction of angular momentum of the particle is fixed

**Question 29. Analogue of mass in rotational motion is**

- Moment of inertia
- Angular momentum
- Gyration
- None of these

**Answer:** 1. Moment of inertia

**Question 30. A constant torque of 3.14 N · m is exerted on a pivoted wheel. If the angular acceleration of the wheel is 4πrad · s ^{-2}, then the moment of inertia of the wheel is**

- 0.25 kg · m
^{2} - 2.5 kg · m
^{2} - 4.5 kg · m
^{2} - 25 kg · m
^{2}

**Answer:** 1. 0.25 kg · m^{2}

**Question 31. A small object of mass m is attached to a light string which passes through a hollow tube. The tube is held by one hand and the string by the other. The object is set into rotation in a circle of radius R and velocity v. The string is then pulled down, shortening the radius of the path of r. What is conserved?**

- Angular momentum
- Linear momentum
- Kinetic energy
- None of these

**Answer:** 1. Angular momentum

**Question 32. The moment of inertia of a disc of radius 5 cm is 0. 02 kg · m². A tangential force of 20 N is applied along the circumference of the disc. The angular acceleration of the disc will be (in unit rad · s ^{-1})**

- 2.5
- 10
- 20
- 50

**Answer:** 4. 50

**Question 33. A body of mass 10 kg and radius 0.5 m is moving in a circular path. The rotational kinetic energy of the body is 32.8 J. Radius of gyration of the body is**

- 0.25 m
- 0.2 m
- 0.5 m
- 0.4 m

**Answer:** 4. 0.4 m

**Question 34. Two discs of the moment of inertia I _{1 }and I_{2} are rotating separately with angular velocities ω_{1} and ω_{2} respectively about a perpendicular axis passing through their centres. If these two rotating discs are connected coaxially then the rotational kinetic energy of the composite system will be**

- \(\frac{I_1 \omega_1+I_2 \omega_2}{2\left(I_1+I_2\right)}\)
- \(\frac{\left(I_1+I_2\right)\left(\omega_1+\omega_2\right)}{2}\)
- \(\frac{\left(I_1 \omega_1+I_2 \omega_2\right)^2}{2\left(I_1+I_2\right)}\)
- \(\frac{\left(I_1+I_2\right)\left(\omega_1+\omega_2\right)^2}{2}\)

**Answer:** 3. \(\frac{\left(I_1 \omega_1+I_2 \omega_2\right)^2}{2\left(I_1+I_2\right)}\)

**In this type of question, more than one option are correct.**

**Question 35. In which of the following cases is the angular momentum conserved?**

- The planet Neptune moves in an elliptical orbit with the sun at one of the foci of the ellipse.
- An electron describes a Sommerfield elliptical orbit around the nucleus.
- An α-particle, approaching a nucleus, is scattered by the force of electrostatic repulsion between the two.
- A boy hurls a stone, tied to a string, in a horizontal circle.

**Answer:**

- The planet Neptune moves in an elliptical orbit with the sun at one of the foci of the ellipse.
- An electron describes a Sommerfield elliptical orbit around the nucleus.
- An α-particle, approaching a nucleus, is scattered by the force of electrostatic repulsion between the two.

**Question 36. A particle of mass m is projected with a velocity v, making an angle of 45° with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection, when it is at its maximum height h, is**

- \(zero\)
- \(\frac{m \nu^3}{4 g \sqrt{2}}\)
- \(\frac{m v^3}{4 \sqrt{2 g}}\)
- \(\frac{m v}{2 g h^3}\)

**Answer:** 2. \(\frac{m \nu^3}{4 g \sqrt{2}}\)

**Question 37. The moment of inertia of a thin square plate ABCD, of uniform thickness about an axis passing through the centre O and perpendicular to the plane is**

- I
_{1}+ I_{2} - I
_{3}+ I_{4} - I
_{1}+ I_{3} - I
_{1}+ I_{2}+ I_{3}+ I_{4}

**Answer: **

- I
_{1}+ I_{2} - I
_{3}+ I_{4}

**Question 38. Choose the correct alternatives**

- For a general rotational motion, angular momentum L and angular velocity ω need not be parallel
- For a rotational motion about a fixed axis, angular momentum L and angular velocity ω are always parallel
- For a general translational motion, momentum \(\vec{p}\) and velocity \(\vec{v}\) are always parallel
- For a general translational motion, acceleration \(\vec{a}\) and velocity \(\vec{v}\) are always parallel

**Answer:**

1. For a general rotational motion, angular momentum L and angular velocity ω need not be parallel

3. For a general translational motion, momentum \(\vec{p}\) and velocity \(\vec{v}\) are always parallel

**Question 39. Net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?**

- The forces may be acting radially from a point on the axis
- Forces may be acting on the axis of rotation
- Forces may be acting parallel to the axis of rotation
- The torque caused by some forces may be equal and opposite to that caused by other forces

**Answer:**

1. The forces may be acting radially from a point on the axis

3. Forces may be acting parallel to the axis of rotation