Question

A rigid body can be hinged about any point on the x-axis. When it is hinged such that the hinge is at x, the moment of inertia is given by $I=2x^2–12x+27$. The x-coordinate of centre of mass is

Answer 1

Moment of inertia is minimum at the centre of mass of the body.
For 'I' to be minimum
$\frac{dI}{dx}=0\Rightarrow \frac{d(2x^2-12x+27)}{dx}=0$
Now differentiate with respect to x,
we get
$\Rightarrow 4x-12=0$
$x=\frac{12}{4}=3$
$x=3,$ hence the x-coordinate of COM is 3.