### Exercise 1

If the polar ice caps completely melt due to warming, then :

#### Solution

The gravitational pull of the Sun passes through COM of the Earth, providing centripetal force required for the rotation of the Earth about it. Thus, there is no external torque on the Earth. It means that the angular momentum of the Earth remains constant.

The melting of ice cap will result in the rise of sea level. From the point of view of MI of the Earth, it means redistribution of mass. The water equivalent of ice moves away from the axis of rotation, which passes through the poles. This results in an increase in the MI of the Earth.

Now, applying law of conservation of angular momentum :

Hence, increase in the MI of the Earth, due to melting of ice, will decrease angular velocity. On the other hand, the duration of a day on the Earth is equal to its time period of rotation, which is given as :

As the Earth rotate slowly (lesser angular speed,

Hence, options (b) and (d) are correct.

### Exercise 2

A circular disk of mass “M” and radius “R” is rotating with angular velocity “ω” about its vertical axis, when two small objects each of mass “m” are gently placed on the rim of the disk. Then, the angular velocity of the ring becomes :

#### Solution

Since no external torque is operating on the disk – objects system, the angular momentum of the system is conserved.

Let “

Here,

Also, the MI of the composite system is :

Putting in the equation of conservation law,

Hence, options (a) is correct.

### Exercise 3

What would be the duration of day, if earth shrinks to half its radius with two – third of its original mass ? Consider motion of the Earth about the Sun along a circular path.

#### Solution

The Earth rotates around the Sun as gravitational pull provides the necessary centripetal force. This force passes through the center of mass of the Earth. As such, gravitational pull does not constitute torque on the Earth. Therefore, we can consider that angular momentum of the Earth remains unchanged. Now, let us use "i" and "f" subscripts to denote initial and final values, then according to law of conservation of angular momentum :

We know that time period of the Earth is :

Hence, its angular velocity is :

Since data on mass and radius are missing, it is not possible to calculate MIs in two cases separately. However, we can find the ratio of MIs :

Putting in the equation,

The new time period of rotation is :

Hence, options (b) is correct.

##### Note:

### Exercise 4

Two disks of moments of inertia

Two rotating disks |
---|

#### Solution

The bodies acquire equal angular velocity due to friction operating between the surfaces in contact. However, friction here is internal to the system of two rotating disks. Thus, there is no external torque on the system and we can employ law of conservation of angular momentum :

Two rotating disks |
---|

Here,

When disks come in tact and rotate about a common axis with equal angular velocity, they acquire common angular velocity, say ”ω”. Since,
Two disks are rotating about a common axis of rotation, the MI of the combination is arithmetic sum of individual MIs. The moment of inertia of the composite system, “

Thus, angular momentum of the system in this situation is :

Putting values in the equation of conservation of angular momentum, we have :

Hence, option (c) is correct.