First slide

Centre of mass(com)

Question

Point masses m₁ and m₂ are placed at the opposite ends of a rigid rod of length l and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position on the rod through which the axis should pass in order that the work required to set the rod rotating with angular velocity $ω_{0}$ be minimum is

Moderate

A

$\frac{M_{2} L}{M_{1} + M_{2}}$
B

$\frac{M_{1} + M_{2}}{M_{2} L}$
C

$\frac{M_{1} L}{M_{1} - M_{2}}$
D

$\frac{M_{1} - M_{2}}{M_{2} L}$

Solution

In order

From work-energy theorem
$W = ΔKE = \frac{1}{2} [m_{1} x^{2} + m_{2} (l - x)^{2}] ω_{0}^{2}$
For minimum work, $\frac{dW}{dx} = 0$
or $2 m_{1} x + 2 m_{2} (l - x) = 0$
$\Rightarrow x = \frac{M_{2} L}{M_{1} + M_{2}}$

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