Question

A train of mass $\mathrm{M}$ is moving on a circular track of radius $\mathrm{R}$ with a constant speed $\mathrm{v}$. The length of the train is half of the perimeter of the track. Find the linear momentum of the train.

Answer 1

Step 1: Given parameters

Mass of train: $\text{Mass}=M$

Length of train $L=\pi R$

Mass per unit length $\lambda =\frac{M}{\pi R}$

Step 2: To Find:

Linear momentum of the train.

Step 3: Calculate the Linear momentum of the train.

Take a small element at an angle $\theta$ on the semicircle which subtends angle $d\theta$.

Let $dm$ be the mass of this small element.

$$\begin{gathered} dm=\lambda dl \\ =\frac{M}{\pi R}dl \end{gathered}$$

$$\begin{gathered} p={\int }_o^{\pi }v\sin \theta dm \\ ={\int }_0^{\pi }\frac{M}{\pi R}\times Rd\theta \left(v\sin \theta \right) \\ =\frac{Mv}{\pi }{\int }_0^{\pi }\sin \theta d\theta \\ =\frac{2Mv}{\pi } \end{gathered}$$