Answer to Question #169590 in Classical Mechanics for Rohan

Question

A particle moves under a central force field. The quantity  [d²/dθ² (1/r)]  is proportional to 1/r. The symbols have standard meaning. The magnitude of the force is proportional to

(a) 1/r²

(b) 1/r

(c) 1/ r³

(d) r

(e) none of these

Answer 1

Given,

$\frac{d^2}{d\theta^2}(\frac{1}{r})=\frac{1}{r}$

Let $\frac{1}{r}=k$

$\frac{d^2k}{d\theta^2}=k$

we can write it as $\frac{d^2k }{k}= d\theta^2$

Now, $\frac{d^2k }{kdt^2}= \frac{d\theta^2}{dt^2}$

Angular acceleration $(\alpha)\propto r\frac{d^2(1/r)}{dt^2}$

Hence, we can conclude that it is proportional to r hence correct answer be option (d)