Point masses `m_(1) and m_(2)` 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 perpendicualr to it. The position of point `P` on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity `omega_(0)` is minimum, is given by :

Explanation

Let the axis of rotation be at a distance x from `m_(1)` Fig `K.E.` of translation `= 0`
`K.E.` of rotation `= (1)/(2) (I_(1) + I_(2))omega_(0)^(2)`
According to work energy principle,
Work done = Increase in energy
`W = (1)/(2) (I_(1) + I_(2)) omega_(0)^(2) = (1)/(2) [m_(1)x^(2) + m_(2) (L - x)^(2)]omega(0)^(2)` ..(i)
For `W` to be minimum,`(dW)/(dx) = 0`
From (i), `(dW)/(dx) = (1)/(2)m_(1)2x + (1)/(2)m_(2) xx 2(L - x) = 0 or m_(1) x - m_(2) (L - x) = 0 or (m_(1) + m_(2)) x = m_(2)L`
`x = m_(2)L//(m_(1) + m_(2))`
This is the position of centre of mass of the ord from `m_(1)`. Hence the required axis should pass through centre of mass of the rod and perpendicular to the length of the rod as shown in Fig.