Question

find the path of center of mass and center of mass velocity if a chain of length L is moving on a circular path of radius R with velocity v.and L=pi R.

Answer 1

Explanations & Calculations

• Consider the figure to get a better understanding.
• Distance to the com (ox) of an arc of radius r is given by, $\small ox = \large \frac{r \sin \alpha}{\alpha}$
• Since $\small L= \pi R$, the chain makes a semicircular arc whose com lies at a distance from the origin as $\small ox = \large \frac{R \sin \frac{\pi}{2}}{\frac{\pi}{2}} = \frac{2R}{\pi}$

• Therefore, as the chain rotates, its center of mass also changes it position being fixed at x which describes a circular path of radius ox.

• Since the chain moves on a circular path at a constant speed, it has an (constant) angular velocity $\small \omega$, which is a constant value about the axis for a given circular motion .
• Therefore,

$\qquad\qquad \begin{aligned} \small \omega &= \small \frac{V}{R}= \frac{V_1}{\text{ox}}\\ \small \frac{V}{R}&= \small \frac{V_1}{\frac{2R}{\pi}}\\ \small V_1 &= \small \frac{2V}{\pi} \end{aligned}$