Question

A satellite moving in a circular path of radius $r$ around earth has a time period $T$. If its radius slightly increases by $4\%$, then percentage change in its time period is:

• A. $1\%$
• B. $6\%$
• C. $3\%$
• D. $9\%$

Answer 1

The correct option is B $6\%$

Given that,

Radius = $r$

Time period = $T$

Now, according to kepler’s third law

$T^2\propto r^3$

$T^2=k{r}^3$

On differentiating

$2T\frac{dT}{dr}=3k{r}^2$

$2T\frac{dT}{dr}=3\frac{T^2}{r}$

Now,

$\frac{dT}{dr}=\frac{3}{2}\times \frac{T}{r}$

Now, the change in time period

$\Delta T=\frac{3}{2}\times \frac{T}{r}\times \Delta r$

If the radius increases by $4$%

So,

$\Delta r=\frac{4}{100}r$

$\Delta r=0.04r$

Now, the time period is

$\Delta T=\frac{3}{2}\times \frac{T}{r}\times 0.04r$

$\Delta T=0.06T$

$\Delta T\%=6\%$

Hence, the change in time period is $6$%