Question

If the earth suddenly contracts to \((\frac{1}{3})^{rd}\) of its present size without change in its mass, the ratio kinetic energy of the earth after and before contraction will be (Earth is assumed to be a rotating sphere about itself)

• A. \(9\)
• B. \(5\)
• C. \(7\)
• D. \(6\)

Hint:

L is conserved, so K is proportional to 1/I.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
There is no external torque on the earth, so angular momentum \(L=I\omega\) is conserved. The rotational kinetic energy is \(K=\dfrac{L^2}{2I}\).

Step 2: Moment of inertia:
For a solid sphere \(I=\tfrac25MR^2\). If \(R\to R/3\) with \(M\) fixed, then \(I'=\dfrac I9\).

Step 3: Energy ratio:
\[ \frac{K'}{K}=\frac{L^2/2I'}{L^2/2I}=\frac{I}{I'}=9 \]

Step 4: Choose:
Option (A). The kinetic energy increases ninefold, with the extra energy coming from the work done by gravity during contraction.

Final Answer:
The kinetic energy becomes 9 times. \[ \boxed{9} \]