Question

A horizontal platform in the shape of a circular disk rotates on a frictionless bearing about a vertical axle through the centre of the disk. The platform has a mass of $150 \,kg$, a radius of $2.0\, m$, and a rotational inertia of $300\, kg \,m ^{2}$ about the axis of rotation. A $60\, kg$ student walks slowly from the rim of the platform toward the centre. If the angular speed of the system is $1.5 \,rad / s$ when the student starts at the rim, what is the angular speed when she is $0.50 m$ from the centre?

• A. 1.2 rad/s
• B. 2.6 rad/s
• C. 1.5 rad/s
• D. 3.6 rad/s

Answer 1

Answer: (B)

The initial rotational inertia of the system is $I_{i}=I_{\text {disk }}$ $+I_{\text {student }}$, where $I_{\text {disk }}=300\, kg \,m ^{2}$ (which, incidentally, does agree with Table $10-2( c ))$ and $I_{\text {student }}=m R^{2}$ where $m=60\, kg$ and $R=2.0 \,m$.
The rotational inertia when the student reaches $r=0.5 \,m$ is $I_{f}=I_{ disk }+m r^{2}$. Angular momentum conservation leads to
$I_{i} \omega_{i}=I_{f} \omega_{f} \Rightarrow \omega_{f}=\omega_{i}\left(\frac{I_{\text {disk }}+m R^{2}}{I_{\text {disk }}+m r^{2}}\right)$
which yields, for $\omega_{i}=1.5 \,rad/s$, a final angular velocity of $\omega_{f}=2.6 \,rad/s$