Bent Rod Moment

Classical Mechanics Level 4

There is a thin uniform rod of mass $M$ in the $xy$ -plane which is initially aligned with the $x$ -axis. The body of the rod begins at the origin $x=0$ and ends at $x = L$ .

Suppose that the portion of the rod up to $x = a$ , where $a < L$ remains aligned with the $x$ -axis, and the portion from $x = a$ to $x = L$ is bent upwards at a right angle so as to be perpendicular to the $xy$ -plane.

The bent rod's moment of inertia with respect to the $z$ -axis can be expressed as ${M a^{2} - \dfrac{\alpha}{\beta} \dfrac{ Ma^{3}}{ L}} .$ If $\alpha$ and $\beta$ are coprime positive integers, determine $\alpha+\beta$ .

The answer is 5.

1 solution

Pranshu Gaba
Nov 16, 2016

Let us draw the diagram to make it clear what the shape of the rod is. The rod is drawn in red and blue. The dashed line is the axis about which we are finding the moment of inertia of the rod. $\si{\kilogram}$

It will be simpler to find the moment of inertia of both parts of the rod separately:

Red:
Its mass is $\frac{a}{L} M$ . The moment of inertia passing through the rod's endpoints and parallel to the $z$ axis is $I_{red} = \left(\frac{a}{L} M \right) \frac{ a^2}{3}$ .

Blue:
Its mass is $\frac{L - a}{L} M$ .The moment of inertia of the blue part passing through its center and parallel to the $z$ axis is 0. Using the parallel axis theorem , its moment of inertia about the given axis is $I_{blue} = \frac{L - a}{L} M a^2$ .

The total moment of inertia of the rod is

$\begin{aligned} I & = I_{red} + I_{blue} \\ & = \frac{Ma^3}{3L} + Ma^2 - \frac{Ma^3}{L} \\ & = Ma^2 - \frac{2}{3} \frac{Ma^3}{L} \end{aligned}$

By comparing it to the expression in the problem, we get $\alpha = 2, \ \beta = 3$ . Therefore $\alpha +\beta = 5$ .

Nice, thanks

Steven Chase - 4 years, 7 months ago

I too have solved the same way. Probably, any one at first will solve like this.
It could be solved by locating the mass center first and solving. That will be long.

Niranjan Khanderia - 4 years, 6 months ago

Bent Rod Moment

The answer is 5.

1 solution

0 pending reports