First slide

Dynamics of rotational motion about fixed axis

Question

A thin horizontal uniform rod AB of mass m and length l can rotate freely about a vertical axis passing through its end A. At a certain moment, the end B starts experiencing a constant force F which is always perpendicular to the original position of the stationary rod and directed in a horizontal plane. The angular velocity of the rod as a function of its rotation angle $θ$ measured relative to the initial position should be

Moderate

A

$\sqrt{\frac{6 F \sin θ}{ml}}$
B

$\sqrt{\frac{2 F \sin θ}{ml}}$
C

$\sqrt{\frac{3 F \sin θ}{ml}}$
D

$\sqrt{\frac{5 F \sin θ}{ml}}$

Solution

Work done by the torque
$∆ W = \int τdθ = \int_{0}^{θ} Fl \cos θ dθ$
$∆ W = Fl \sin θ$
Now using work energy theorem,
$∆ W = ∆ k$
$∴ Flsinθ = [\frac{1}{2} (\frac{{ml}^{2}}{3}) ω^{2} - 0]$
Which gives, $ω = \sqrt{\frac{6 F \sin θ}{ml}}$

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