Question

Suddenly the earth contracts its radius is half, normally it take time 24hrs to complete it's rotation, then how long the earth will take to complete its rotation when its contract its radius???​

Answer 1

Answer:

We know that reduction in Earth’s size implies the reduction in its radius, so, here the radius is reduced to half. Find the moment of inertia of earth before and after contraction keeping in mind that the mass remains constant. Now, you could substitute them and the expression of angular velocities before and after contraction in the law of conservation of angular momentum and thus get the time period of one rotation of earth, that is, the duration of a day.

Formula used:

Moment of inertia of a solid sphere,

I=25MR2I=25MR2

Expression for angular velocity,

ω=2πTω=2πT

Law of conservation of angular momentum,

I1ω1=I2ω2I1ω1=I2ω2

Complete step-by-step answer:

We are given that the Earth is contracted to half its present size. Change in size implies change in radius, so, here the present radius of earth (R1)(R1) is reduced by half. That is, after contraction the radius becomes,

R2=R12R2=R12 ………………………………. (1)

But during the contraction, the mass remains constant. That is,

M1=M2=MM1=M2=M ……………………………… (2)

Due to the above mentioned contraction, the angular velocity of Earth’s rotation changes. Also, we know that angular velocity is given by,

ω=2πTω=2πT

Angular velocity before contraction,

⇒ω1=2πT1⇒ω1=2πT1 ………………………… (3)

Angular velocity after contraction,

⇒ω2=2πT2⇒ω2=2πT2 ……………………………….. (4)

There is another quantity that is undergoing change due to contraction of earth – moment of inertia.

Moment of inertia of a solid sphere is given by,

I=25MR2I=25MR2

Moment of inertia before contraction,

I1=25MR12I1=25MR12 …………………………. (5)

Moment of inertia after contraction,

I2=25MR22=25M(R12)2=I14I2=25MR22=25M(R12)2=I14 ………………………………. (6)

Now, by the law of conservation of angular momentum,

dLdt=0dLdt=0

⇒L=Iω⇒L=Iω= constant

⇒I1ω1=I2ω2⇒I1ω1=I2ω2 …………………………….. (7)

Substituting (3), (4) and (6) in (7), we get,

I1(2πT1)=(I14)(2πT2)I1(2πT1)=(I14)(2πT2)

⇒T2=T14⇒T2=T14 ………………….. (8)

But we know the present time period rotation of our planet Earth as 24 hours. Therefore,

T1=24HrsT1=24Hrs …………… (9)

Substituting (9) in equation (8), we get,

T2=244T2=244

⇒T2=6Hrs⇒T2=6Hrs

Time period of rotation of Earth is what we call a day. So, for an Earth that is contracted to half its present size with mass constant, the duration of a day will be just 6hrs.