Ncert Exercises & Solutions

Question 7. 13. (a) A child stands at the center of a turntable with his arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value? Assume that the turntable rotates without friction.

(b) Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

100 rev/min
Initial angular velocity, ω₁= 40 rev/min
Final angular velocity = ω₂
The moment of inertia of the boy with stretched hands = I₁

The moment of inertia of the boy with folded hands = I₂
The two moments of inertia are related as:

I_{2} = \frac{2}{5} I_{1}

Since no external force acts on the boy, the angular momentum L is a constant.
Hence, for the two situations, we can write:

I_{2} ω_{2} = I_{1} ω_{1} ω_{2} = \frac{I_{1}}{I_{2}} ω_{1} = \frac{I_{1}}{\frac{2}{5} I_{1}} \times 40 = \frac{5}{2} \times 40 = 100 rev / \min

(b) Final K.E. = 2.5 Initial K.E.
Final kinetic rotation,

E_{F} = \frac{1}{2} I_{2} {ω_{2}}^{2}

Initial kinetic rotation,

E_{I} = \frac{1}{2} I_{1} {ω_{1}}^{2}

\frac{E_{F}}{E_{I}} = \frac{\frac{1}{2} I_{2} {ω_{2}}^{2}}{\frac{1}{2} I_{1} {ω_{1}}^{2}} = \frac{2}{5} \frac{I_{1}}{I_{1}} \frac{{(100)}^{2}}{{(40)}^{2}} = \frac{2}{5} \times \frac{100 \times 100}{40 \times 40} = \frac{5}{2} = 2.5 ∴ E_{F} = 2.5 E_{I}

The increase in the rotational kinetic energy is attributed to the internal energy of the boy.

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