Question

Suppose several particles are in concentric circular orbits under the influence of some centripetal component of force, the magnitude of which depends on distance from the center of the motion. Each particle has a different orbital radius r and a corresponding period T.How does the force depend on r if T is proportional to sqrt(r) ?
How does the force depend on r if T is proportional to r^3/2 ?
How does the force depend on r if T is independent of r ?

Answer 1

Explanation:

The centripetal force acting on a particle is given by :

$F=\dfrac{mv^2}{r}$

Since, $v=r\omega$

$F=mr\omega^2$

$F\propto r\omega^2$

Case 1.

If $T\propto \sqrt{r}$

Since, $T=\dfrac{2\pi}{\omega}$

So, $\omega\propto \dfrac{1}{\sqrt{r} }$

$\omega^2\propto \dfrac{1}{r}$

So, the force becomes,

$F\propto r\times \dfrac{1}{r}$

F is independent of r

Case 2.

If $T\propto r^{3/2}$

$\omega\propto \dfrac{1}{r^{3/2}}$

$\omega^2 \propto \dfrac{1}{r^3}$

So, $F\propto r\times \dfrac{1}{r^2}$

So, the force is inversely proportional to the square of radius.

Case 3.

If T is independent of r, the force will be directly proportional to the radius of orbit.

Hence, this is the required solution.