Question

Two point masses $6\text{kg}$ each are connected by a thin massless rod of length $10\text{m}$. Where should the axis of rotation of the system perpendicular to rod, in order to have the minimum moment of inertia?

• A. $\text{3 m}$ away from $a$
• B. $\text{4 m}$ away from $a$
• C. At the mid point of the rod.
• D. None of these

Answer 1

The correct option is C At the mid point of the rod.


Let axis of rotation be at $x$ distance away from particle $a$

we know,
Moment of inertia of system
$I=I_a+I_b=m{r}_a^2+m{r}_b^2=6\times x^2+6\times (10-x{)}^2$

To minimise $I$
$\frac{dI}{dx}=0$ and $\frac{d^{2}I}{d{x}^2}\ge 0$

$\frac{dI}{dx}=12x-12(10-x)=0$
$\Rightarrow x=5\text{m}$
And
$\frac{d^{2}I}{d{x}^2}=24>0$

$\therefore$ Moment of inertia of system will be minimum when axis of rotation will be in between $a$ and $b$ mass and at the mid point of the rod.