Question 66: Centre of Mass and Rotational Motion

Question

A thin but rigid semicircular wire frame of radius $$r$$ is hinged at $$O$$ and can rotate in its own vertical plane. A smooth peg $$P$$ starts from $$O$$ and moves horizontally with constant speed $$2 v_0$$ lifting the frame upward as shown in figure. Find the angular velocity $$\omega$$ of the frame when its diameter makes an angle of $$60^{\circ}$$ with vertical.

Accepted Answer

$$\begin{aligned}
\frac{x}{\sin 	heta} & =\frac{2 r}{\sin 90^{\circ}} \Rightarrow x=2 r \sin 	heta \
	herefore \quad \frac{d x}{d t} & =2 r \cos 	heta \frac{d 	heta}{d t} \
\frac{d 	heta}{d t} & =\frac{d x / d t}{2 r \cos 	heta}=\frac{2 v_0}{2 r \cos 60^{\circ}}=\frac{2 v_0}{r}
\end{aligned}$$

Answer

$$v_0 / r$$

Answer

$$v_0 / 2 r$$

Answer

$$2 v_0 / r$$

Answer

$$v_0 r$$

Centre of Mass and Rotational Motion

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