Question

Consider an attractive force which is central but is inversely proportional to the first power of distance. If a particle is in circular orbit, under such a force, which of the following statements are correct?

• A. The speed is directly proportional to the square root of orbital radius
• B. The speed is independent of radius
• C. The period is independent of radius
• D. The period is directly proportional to radius

Answer 1

Answer: (B), (D)

The correct options are
B The speed is independent of radius
D The period is directly proportional to radius

Let the Force be, $F=\frac{k}{r}$

where $k$ is some proportionality constant. Since the force is centripetal in this case:

$\frac{m{v}^2}{r}=\frac{k}{r}\Rightarrow v=\sqrt{\frac{k}{m}}$

$\therefore$ Speed is independent of radius.

angular speed $w=\frac{v^2}{r}=\frac{k}{rm}$

time period $T=\frac{2\pi }{w}=\frac{2\pi rm}{k}$

$\therefore$ time period is directly proportional to radius.