Question 480: Centre of Mass and Rotational Motion

Question

A man standing on a turn-table is rotating at a certain angular frequency with his arms outstretched. He suddenly folds his arms. If his moment of inertia with folded arms is $75 \%$ of that with outstretched arms, then his rotational kinetic energy will

Accepted Answer

Conservation of Angular Momentum
When the man folds his arms, his total spin (angular momentum) does not change because no outside force acts on him.
Mathematically,

$$  \begin{aligned}  & I_1 \omega_1 = I_2 \omega_2 \ & 	ext{We know: } I_2 = 75\%\, 	ext{of}\,I_1 = \frac{3}{4} I_1  \end{aligned}  $$
So,

$$  I_1 \omega_1 = \frac{3}{4}I_1\omega_2  $$
Divide both sides by $I_1$:

$$  \omega_1 = \frac{3}{4} \omega_2  $$
So,

$$  \omega_2 = \frac{4}{3} \omega_1  $$
Rotational Kinetic Energy
When his arms are outstretched:

$$  K.E_1 = \frac{1}{2} I_1 \omega_1^2  $$
When his arms are folded:

$$  K.E_2 = \frac{1}{2} I_2 \omega_2^2  $$
Substitute $I_2 = \frac{3}{4}I_1$ and $\omega_2 = \frac{4}{3}\omega_1$:

$$  K.E_2 = \frac{1}{2} (\frac{3}{4}I_1) (\frac{4}{3}\omega_1)^2  $$
Now, $(\frac{4}{3})^2 = \frac{16}{9}$, so:

$$  K.E_2 = \frac{1}{2} 	imes \frac{3}{4} I_1 	imes \frac{16}{9} \omega_1^2  $$
Simplify:

$$  K.E_2 = \frac{1}{2} I_1 \omega_1^2 	imes \frac{3}{4} 	imes \frac{16}{9} $$
Multiply the fractions:

$$  \frac{3}{4} 	imes \frac{16}{9} = \frac{48}{36} = \frac{4}{3}  $$
So,

$$  K.E_2 = \frac{4}{3} K.E_1  $$
Percentage Increase in Kinetic Energy
Change in energy:

$$  	ext{Increase} = K.E_2 - K.E_1 = \frac{4}{3}K.E_1 - K.E_1 = \frac{1}{3}K.E_1  $$
Percentage increase:

$$  	ext{Percentage} = \frac{1}{3} 	imes 100 = 33.3\%  $$
Final Answer
When the man folds his arms, his rotational kinetic energy increases by 33.3%.

Answer

increase by $33.3 \%$

Answer

decrease by $33.3 \%$

Answer

increase by $25.0 \%$

Answer

decrease by $25.0 \%$

Centre of Mass and Rotational Motion

A man standing on a turn-table is rotating at a certain angular frequency with his arms outstretched. He suddenly folds his arms. If his moment of inertia with folded arms is 75 \% of that with outstretched arms, then his rotational kinetic energy will

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