First slide

Vertical circular motion

Question

A hemispherical bowl of radius R is set rotating about its axis of symmetry. A small body kept in the bowl rotates with the bowl without slipping on its surface. If radius through the body makes with the axis an angle $θ$ . $θ$ and assuming that the surface of the bowl is smooth, the angular velocity with which bowl is rotating is given by

Moderate

A

$\frac{R}{gcos θ}$
B

$\sqrt{(\frac{R}{gcos θ})}$
C

$\sqrt{(\frac{g}{Rcos θ})}$
D

$\frac{g}{Rcos θ}$

Solution

The different forces on the small body are shown in fig.
From figure $\sin θ = (r / R)$ ……………..(i)
Now, $Ncos θ = mg$ ……………………(ii)
and $Nsin θ = {mrω}^{2}$ …………………….(iii)
Dividing eq. (iii) by eq. (ii), we get
$\tan θ = \frac{{rω}^{2}}{g} = \frac{(Rsin θ) ω^{2}}{g}$
or $\frac{\sin θ}{\cos θ} = \frac{(Rsin θ) ω^{2}}{g}$
or $ω^{2} = \frac{g}{Rcos θ} or ω = \sqrt{(\frac{g}{Rcos θ})}$

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