Question

A satellite moving on a circular path of radius $r$ around Earth has a time period $T$. If its radius slightly increases by $\Delta r$, the change in its time period is

• A. $\frac{3}{2}\left(\frac{T}{r}\right)\Delta r$
• B. $\left(\frac{T}{r}\right)\Delta r$
• C. $\frac{3}{2}\left(\frac{T^2}{r^2}\right)\Delta r$
• D. none of these

Answer 1

Answer: (A)

$\frac{3}{2}\left(\frac{T}{r}\right)\Delta r$

Time period $T^2=\frac{4{\pi }^2}{GM}{r}^3$
Differentiating with respect to $r$
$2T\Delta T=\frac{4{\pi }^2}{GM}\times 3r^2\Delta r$
But $\frac{T^2}{r}=\frac{4{\pi }^2}{GM}{r}^2$
$\Rightarrow 2T\Delta T=T^2\frac{3}{r}\Delta r$
$\Rightarrow \Delta T=\frac{3}{2}\frac{T}{r}\Delta r$