Question

A particle moves in a circular orbit of radius $r$ under a central attractive force $F=-\frac{k}{r},k$ is constant. The time period of its motion shall be proportional to

• A. $r^{1/2}$
• B. $r$
• C. $r^{3/2}$
• D. $r^{2/3}$

Answer 1

Answer: (B)

$r$

Answer 2

Central attractive force
$F=-\frac{k}{r} m r \omega^{2}=-\frac{k}{r}$
$m r\left(\frac{2 \pi}{T}\right)^{2}=-\frac{k}{T} \frac{m r 4 \pi^{2}}{T^{2}}$
$=-\frac{k}{T} T^{2}=\frac{m r^{2} 4 \pi^{2}}{k} T^{2} \propto r^{2}$
or $T \propto r$