Q: Two point masses of 3.0 kg and 0.7 kg are fixed at the ends of a rod of length 1.4 m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of

(a) 0.42 m from mass of 0.3 kg

(b) 0.70 m from mass of 0.7 kg

(c)0.98 m from mass of 0.3 kg

(d) 0.98 m from mass of 0.7 kg

Ans: (c)

Sol: Work done $\large W = \frac{1}{2}I\omega^2 $

Let x = distance of mass 0.3 kg from the centre of mass .

Moment of Inertia , $\large I = 0.3 \times x^2 + 0.7 \times (1.4-x)^2$

For Work to be minimum , the moment of Inertia I should be minimum .

$\large \frac{dI}{dx} = 0$

$\large 0.3 \times 2x + 0.7 \times 2(1.4-x)(-1) = 0$

0.3 x = 0.7 (1.4-x)

$\large x = \frac{0.7 \times 1.4}{0.3 + 0.7}$

x = 0.98 m