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Q1. What is the purpose of attaching a flywheel on a space craft?

Solution

A flywheel is attached on a space craft for controlling the orientation of the space craft. When neither the flywheel nor the space craft is turning then the total angular momentum is zero. To change the orientation of the spacecraft, the flywheel is made to rotate. The spacecraft starts rotating in the opposite sense to maintain the angular momentum of the system zero.When the flywheel is brought to rest, the space craft also stops rotating.But the orientation of space craft changes. 

Q2. Convex and concave lenses of same mass and radius rotate about an axis passing through their centre and perpendicular to the plane. Which one will have greater moment of inertia?

Solution

The concave lens is thin at the centre and thick at the edges. As, more mass is concentrated away from the axis of rotation in case of concave lens, it will have more moment of inertia.

Q3. What is the physical significance of moment of inertia?

Solution

Moment of inertia plays the same role as is played by mass in translatory motion. In translatory motion, mass is a measure of inertia, therefore, moment of inertia is a measure of rotational inertia in rotatory motion.

Q4. What is the unit of angular momentum? What is its dimensional formula?

Solution

Kg m²s^-1 is the unit of angular momentum. Its dimensional formula is [M¹ L² T^-1].

Q5. A projectile projected at a certain angle with the horizontal hits the ground at P. However, if this particle explodes at the highest point into two pieces of equal mass. One piece falls to the ground vertically downwards at Q, while the other travels horizontally and hit the ground at R. What is the distance of R from P?

Solution

In the explosion, only the internal forces are involved and internal forces can not change the motion of centre of mass, So after explosion will takes place, the centre of mass continue to move along the same parabolic path. Since the two fragments are of equal mass, therefore, P would be the mid way between Q and R.

Q6. There are two men standing facing each other on two boats floating on still water at a short distance apart. A rope is held at its ends by both. Explain why the two boats meet at same point when each man pulls the rope separately or both pull together? Will the time taken be different in the two cases? (Neglect Friction)

Solution

As the forces exerted by the two men in boats are the internal forces of the system, therefore, the position of centre of mass remains unchanged. The two boats always meet at the same point which is the centre of mass of the system. It remains unchanged whether both the men pull together or each man pulls separately. However, the boats shall take shorter time to meet when both the men pull together.

Q7. Two bodies A and B are attracted towards each other due to gravitation. Given that A is much heavier than B. what will you say about the relative motion of the centre of mass?

Solution

The two bodies will move towards their common centre of mass. But the location of centre of mass will remain unchanged. i.e Centre of mass remains at rest with respect to A as well as B.

Q8. What is the centre of mass of a rigid body?

Solution

It is the point in or near an object at which the whole mass of the object may be considered to be concentrated. Centre of mass is inside the object if it is a solid symmetrical homogeneous object such as a solid sphere or cube. Its centre of mass is at its geometrical centre. 

Q9. What do you mean by centre of mass frame of reference?

Solution

A frame of reference carried by the centre of mass of an isolated system of particles (i.e a system not subjected to any external forces) is called centre of mass frame of reference. In this frame of reference (i)              Position vector of centre of mass is zero. (ii)             Velocity and hence momentum of centre of mass is also zero.

Q10. A planet revolves around a massive star in a highly elliptical orbit. Is its angular momentum constant over the entire orbit?

Solution

Yes, angular momentum of the planet is constant over the entire orbit. This is because revolution of planet around the star is under the effect of gravitational force between the star and the planet. This is a radial force whose torque is zero. Therefore, angular momentum of the planet is a constant (vector), whatever is the nature of the orbit.

Q11. Torque is a scalar or a vector quantity? If it is a vector quantity where does the vector representing torque point?

Solution

Torque is a vector quantity. The vector representing torque points perpendicular to the plane of motion.

Q12. What do you mean by Torque?

Solution

Torque is the rotational analogue of force which gives us the turning effect of the force about a fixed point or axis. It is measured by the product of the magnitude of force and perpendicular distance of the line of action of force from the axis of rotation.

Q13. Describe the significance of the concept of centre of mass.

Solution

Centre of mass of a system is a point where the total mass of the system is supposed to be concentrated and all the external forces act there. So, the description of motion becomes easier in terms of centre of mass of the system. Thus centre of mass enables us

• To apply Newton's law of motion to extended bodies without difficulty
• To apply law of conservation of momentum/energy to a single particle rather than individual particles.

Q14. What will be the position of centre of mass of a two particle system with unequal masses?

Solution

The centre of mass will lie near to the massive particle.

Q15. Explain the motion of Earth-Moon system around the sun in terms of centre of mass concept.

Solution

The moon revolves around the earth in a circular orbit and the earth goes around the sun in an elliptical orbit. Actually, the Earth-Moon system revolves about their common centre of mass in elliptical path. As the mass of earth is very large as compared to that of the moon, the centre of mass of the Earth-Moon system lies within the earth and it appears that the moon revolves around the earth but the reality is that both revolve around their common centre of mass.

Q16. Which component of torque will affect a rigid body?

Solution

The axis is fixed, only those components of torques, which are along the direction of the fixed axis, need to be considered in our discussion. Only these components can cause the body to rotate about the axis. A component of the torque perpendicular to the axis of rotation will tend to turn the axis from its position.

Q17. Two men stand facing each other on two boats floating on still water at a short distance apart. Both hold a rope at its ends. The two boats are found to meet always at same point, whether each man pulls separately or both pull together, why? Will the time taken be different in the two cases? Neglect friction.

Solution

The men on the two boats floating on the still water constitute a single system and the forces applied by them on each other are internal forces. When the man on the each boat pulls separately or both pull together, the centre of mass of the syatem of the boats remains fixed. It is because, centre of mass moves under the effect of external force only. Therefore, the two boats always meet at a fixed point, which is the centre of mass of the system. It does not matter, whether each man pulls separately or both pull together. However, the time taken in the two cases will not be the same. The boats will take shorter time to meet, when both the men pull together.

Q18. A rocket is out in free space shooting out a stream of exhaust gases and picking up speed in the opposite direction. What happens to centre of mass of all the matter, that which is ejected and that, which is left in the rocket?

Solution

A rocket moving in the free space represents an isolated system, consisting of the ejected matter and that which is left in the rocket. Since no external force is acting on the rocket system, the acceleration of its centre of mass of all the matter remains constant.

Q19. A cap of pen can be easily opened with the help of two fingers than one finger. Why?

Solution

The force applied using two fingers constitute a couple or torque and the cap of pen can be turned easily. But with one finger, only a single force can be applied.

Q20. If Angular momentum is conserved in a system whose moment of inertia is decreased, will its rotational kinetic energy be also conserved?Explain.

Solution

Here L=Iω = constant Kinetic energy of rotation, K = (1/2)I ω² When moment of inertia (I) decreases, kinetic energy of rotation (K)  increases.Thus kinetic energy of rotation is not conserved. 

Q21. It is difficult to open heavy door from a point near the hinges. Why?

Solution

Yes it is difficult to open heavy door from a point near the hinges as that will create less turning effect making the action difficult. If we open the same door from a point farther from the hinges (at the maximum distance from the hinges) then it will create more turning effect hence making the action easier.

Q22. What is the dimensional formula of Torque? Also write its S.I units?

Solution

Torque= Force x Perpendicular distance Dimensional formula of torque = [M ¹ L ¹ T ^-2] [L]                                             = [M ¹ L ² T ^-2] S.I unit of torque is Nm which is equivalent to joule.

Q23. What will be the position of centre of mass in case of a plane lamina of the shape of a rectangle?

Solution

It will be at the point of intersection of two diagonals.

Q24. What do you mean by radius of gyration? Is radius of gyration a constant quantity?

Solution

For any body rotating about a given axis, it is possible to find a radial distance from the axis where, if the whole mass of the body is supposed to be concentrated, its moment of inertia remain unchanged. This radial distance is called radius of gyration.   Radius of gyration is not a constant quantity but its value changes with change of location of axis of rotation.Its value depends on shape and size of the body, position and configuration of the axis of rotation and also on distribution of mass of the body with respect to the axis of rotation.

Q25. What do you understand by mechanical equilibrium of a rigid body?

Solution

For mechanical equilibrium of a rigid body, two conditions need to be satisfied: 1. Translational equilibrium: The net external force or the vector sum of all the external forces acting on the body must be zero.i.e  2.Rotational equilibrium :The net external torque or the vector sum of all the torques acting on the body is zero. i.e 

Q26. Explain why in a radioactive decay the heavy fragment moves with smaller velocity and light fragment moves with a larger velocity in opposite direction.

Solution

During radioactive decay, when a radioactive nucleus initially at rest decays, the fragments fly off in different directions obeying the principles of momentum and energy conservation. As the decay occurs simultaneously, no external forces are involved. The momentum of the system is conserved. As the parent nucleus is initially at rest, its momentum before decay is zero. The fragments fly off in different direction with different velocities so that the vector sum of linear moment of all fragments is zero. The momentum of the centre of mass of all fragments must also be zero after decay. Hence, the centre of mass of entire system of fragments shall continue to be at rest after decay. The decay of a parent nucleus results in one heavy and one light fragment. Hence, heavy nuclei move with smaller velocity in opposite directions as per the conservation of momentum.

Q27. Which principle or law is responsible for high angular speed of inner layers of tornado?

Solution

The principle of angular momentum is responsible for high angular speed of inner layers of tornado.

Q28. Explain briefly the motion of centre of mass?

Solution

In the absence of external force, the centre of mass CM of a system either remains stationary or moves with uniform velocity. When external force is applied on the system, the CM moves with acceleration.

Q29. If a ball or a wheel is made to slide down an inclined plane, what kind of motion can it exhibit?

Solution

It has both translational and rotational motion simultaneously.

Q30. If a body is rotating, is it necessary being acted upon by an external Torque?

Solution

No, Torque is required only for producing angular acceleration. For uniform rotation, no torque is needed.

Q31. Radius of gyration is a constant quantity.

Solution

No. Radius of gyration depends on axis of rotation and distribution of mass.

Q32. Moment of inertia in rotational motion is an analogue of a physical quantity in linear motion. Name the quantity.

Solution

Moment of inertia is analogous to Mass.

Q33. Will it be correct to write the unit of torque as Joule ?

Solution

No. The correct unit of torque is newton-metre, and not joule. This is done to create a difference between the scalar nature of work and the vector nature of torque.

Q34. What do you understand by a rigid body?

Solution

A rigid body is a solid body of finite size which can not be deformed and the distance between any two points on the rigid body does not change much with time. Also, there is no migration of mass within a rigid body. The whole body moves as one, under the effect of external forces.

Q35. Give four examples of a rigid body in pure rotation?

Solution

Ceiling fan, Potters' wheel,  table fan or pedestal fan, and a merry-go-around exhibit pure rotation about an axis.

Q36. A body X of mass M while falling vertically downwards under the effect of gravity break in to two parts A and B having masses M/5 and 4M/5 respectively. How does the centre of mass of bodies A and B taken together shift compared to that of body X?

Solution

The body X break in to A and B because of the internal forces. There was no external force acting on the system. Hence the centre of mass of the two parts A and B will not shift, and follow the original path.

Q37. What is the physical meaning of Torque?

Solution

Torque τ is defined as a quantity in rotational motion, which when multiplied by a small angular displacement gives us work done in rotational motion. This quantity corresponds to force in linear motion, which when multiplied by a small linear displacement gives us work done in linear motion.

Q38. When will you say that a body is in partial equilibrium?

Solution

A body is said to be in partial equilibrium when it is in translation equilibrium but not in rotational equilibrium, or it is in rotational equilibrium and not in translational equilibrium.

Q39. Show that angular momentum of a free particle can be made zero by a suitable choice of origin.

Solution

Choose the origin on the line of motion of the particle. With the choice, the momentum vector  is parallel to the position vector .

Q40. How moment of inertia of a body is different from its mass?

Solution

Mass of a body is a constant quantity, whereas moment of inertia of a body depends upon its mass, size, shape, position and orientation, axis of rotation, distribution of mass.