Question

Two small satellites move in circular orbits around the earth, the distances r and $r+\Delta$r from the centre of the earth. Their time period of rotation are $T$ and $T+$$\Delta$T. ($\Delta$r << r, $\Delta$T << T)

• A. $\Delta T=\frac{3}{2}T\frac{\Delta r}{r}$
• B. $\Delta T=-\frac{3}{2}T\frac{\Delta r}{r}$
• C. $\Delta T=\frac{2}{3}T\frac{\Delta r}{r}$
• D. $\Delta T=T\frac{\Delta r}{r}$

Answer 1

The correct option is A $\Delta T=\frac{3}{2}T\frac{\Delta r}{r}$

As $T^2\propto r^3\Rightarrow T^2=c{r}^3-\left(1\right)$, for some constant $c$.

So, $(T+\Delta T{)}^2=c(r+\Delta r{)}^3$.

As, $\Delta T<T;\Delta r<r$

So, $T^2+2T\Delta T=c(r^3+3r^2\Delta r)-\left(2\right)$

From $\left(1\right)and\left(2\right)$

$2T\Delta T=c3r^2\Delta r$

Dividing by $\left(1\right)$,

$2\Delta T/T=3\Delta r/r$

$\Rightarrow \Delta T=3T\Delta r/2r$