KCET · Physics · Rotational Motion
If the earth were to suddenly contract to half of its present radius, what would be the duration of the day?
- A \(6\) h
- B \(18\) h
- C \(24\) h
- D \(30\) h
Answer & Solution
Correct Answer
(A) \(6\) h
Step-by-step Solution
Detailed explanation
By the principle of conservation of angular momentum, the angular momentum of the Earth remains constant since no external torque acts on it.
\(I_1 \omega_1 = I_2 \omega_2\)
\(\dfrac{2}{5} M R^2 \left(\dfrac{2\pi}{T_1}\right) = \dfrac{2}{5} M \left(\dfrac{R}{2}\right)^2 \left(\dfrac{2\pi}{T_2}\right)\)
\(\dfrac{R^2}{T_1} = \dfrac{R^2}{4 T_2}\)
\(T_2 = \dfrac{T_1}{4}\)
Given \(T_1 = 24\) h, we get:
\(T_2 = \dfrac{24}{4} = 6\) h
Answer: \(6\) h
\(I_1 \omega_1 = I_2 \omega_2\)
\(\dfrac{2}{5} M R^2 \left(\dfrac{2\pi}{T_1}\right) = \dfrac{2}{5} M \left(\dfrac{R}{2}\right)^2 \left(\dfrac{2\pi}{T_2}\right)\)
\(\dfrac{R^2}{T_1} = \dfrac{R^2}{4 T_2}\)
\(T_2 = \dfrac{T_1}{4}\)
Given \(T_1 = 24\) h, we get:
\(T_2 = \dfrac{24}{4} = 6\) h
Answer: \(6\) h
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