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  4. The relative uncertainty in the period of a satellite orbiting around the earth is 10-2. If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is :

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Q. The relative uncertainty in the period of a satellite orbiting around the earth is $10^{-2}$. If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is :

JEE MainJEE Main 2018Physical World, Units and Measurements

Solution:

Time period of revolutions of a satellite revolving around Earth of mass $M$ in orbit of radius $r$ is given by
image
$T=2 \pi \sqrt{\frac{r^{3}}{G M}}$
$\Rightarrow T^{2}=\frac{4 \pi^{2} r^{3}}{G M}$
Taking log of both sides, we get
$\log T^{2}=\log \frac{4 \pi^{2} r^{3}}{G M}=\log \frac{4 \pi^{2} r^{3}}{G}-\log (M)$
$\Rightarrow 2 \log T=\log k-\log M$
$\Rightarrow 2 \frac{\Delta T}{T}=\frac{\Delta M}{M}$
$\Rightarrow \frac{\Delta M}{M}=2 \times 10^{-2}$
($\because$ Given relative uncertainty in period, $\frac{\Delta T}{T}=10^{-2}$ )