First slide

Rigid body doing transational plus rotational motion

Question

In the arrangement shown, the end B of the rod of length ‘l’ is made to move with constant velocity V. Then velocity of its centre of mass when $θ = 30^{0}$ is

Moderate

A

$\frac{v}{\sqrt{2}}$
B

$\sqrt{2} v$
C

$\frac{v}{2 \sqrt{2}}$
D

v

Solution

O is the position of instantaneous centre of rotation. Then entire rod is rotating about O with angular velocity $ω$ at that instant
where $ω = \frac{v_{B}}{OB} = \frac{v}{l \sin θ}$
Now ${(OA)}^{2} + {(OB)}^{2} = 2 . ({AC}^{2} + {OC}^{2})$
$\Rightarrow l^{2} \cos^{2} θ + l^{2} \sin^{2} θ = 2 (\frac{l^{2}}{4} + {OC}^{2})$
$\Rightarrow {OC}^{2} = \frac{l^{2}}{2} - \frac{l^{2}}{4} = \frac{l^{2}}{4} \Rightarrow OC = \frac{l}{2}$
$∴ v_{C} = ω . (OC) = \frac{v}{l \sin θ} \times \frac{l}{2} = \frac{v}{2 \sin θ}$
When $θ = 30^{0}, v_{C} = \frac{v}{2 \sin 30^{0}} = v$

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