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3-2.Motion in Plane
hard
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 $v_0$, 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 the vertical :
A
$v_0 / r$
B
$v_0 / 2 r$
C
$2 v_0 / r$
D
$v_0 r$
Solution
(a)
$\frac{x}{\sin 2 \theta}=\frac{r}{\sin (90-\theta)}$
$\Rightarrow x=2 r \sin \theta$
$\therefore \frac{d x}{d t}=2 r \cos \theta \times \frac{d \theta}{d t}$
$\frac{d \theta}{d t}=\frac{d x / d t}{2 r \cos \theta}=\frac{v_0}{2 r \cos 60^{\circ}}=\frac{v_0}{r}$
Standard 11
Physics
