Question

Two point masses A of mass M and B of mass 4M are fixed at the ends of a rod of length l and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The work required for rotating the rod will be minimum when the distance of axis of rotation from the mass A is at

• A. $\frac{l}{5}$
• B. $\frac{4}{5}l$
• C. $\frac{3}{5}l$
• D. $\frac{2}{5}l$

Answer 1

The correct option is B

$\frac{4}{5}l$

$W=\frac{1}{2}I{\omega }^2$

Let x is the distance of CM from A

$I=M{x}^2+4M(l-x{)}^2$

And if I is minimum, W will be minimum

$\therefore \frac{dI}{dx}=2Mx+4M\times 2(l-x)\times (-1)$

$=2Mx-8M(l-x)$

$\frac{dI}{dx}=10Mx-8Ml$

$\therefore 10Mx-8Ml=0$

$x=\frac{4}{5}l$
Which is the position of center mass of this system.