A low-mass rod of length $0.30m$has a metal ball of mass $1.4kg$at each end. The center of the rod is located at the origin, and the rod rotates in the $yz$plane about its center. The rod rotates clockwise around its axis when viewed form a point on the $+x$axis, looking towards the origin. The rod makes one complete rotation every$0.5s$ (a) what is the moment of inertia of the object (rod plus two balls)? (b) what is the rotational angular momentum of the object? (c) What is the rotational kinetic energy of the object?

The moment of inertia is a quantitative measure of a body's rotational inertia, or its resistance to having its speed of rotation about an axis changed by the application of a torque.

The rotating analogue of linear momentum is angular momentum (also known as moment of momentum or rotational momentum). Because it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics.

Length of a low-mass rod,$L=0.30m$

Mass of each metal ball,$m=1.7kg$

Period of rotation,$T=0.5s$

Angular speed of the rod, $\omega =1rev/0.5s$

$=\left(1rev/0.5s\right)\left(\frac{2\pi rad}{1rev}\right)$

$=12.57rad/s$

Because the rod has a low mass, the contribution of the rod's moment of inertia to the overall moment of inertia of the object can be ignored. There are two metal balls on each end of the rod.

its end at a distance $\frac{L}{2}$from the center of the rod. So, the moment of inertia of the object is sum of the moment of inertias of each object about the center of rod. Hence, the moment of inertia of the object is

$$\begin{gathered} I_{total}=I_{ball}+I_{ball} \\ =m{\left(\frac{L}{2}\right)}^2+m{\left(\frac{L}{2}\right)}^2 \\ =2m\left(\frac{L^2}{4}\right) \end{gathered}$$

$I_{total}=\frac{1}{2}m{L}^2....\left(2\right)$

On substituting the known values in the equation (2) , the moment of inertia of the object can be calculated as

$$\begin{gathered} I_{total}=\frac{1}{2}\left(1.7kg\right){\left(0.30m\right)}^2 \\ =0.0765kg·m^2 \end{gathered}$$