Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsGravitation3 Marks
Question
Viscous force increase the velocity of a satellite. Discuss.
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Answer
Imagine a satellite of mass m moving with a velocity v in an orbit of radius r around a planet of mass M. PE of the satellite, $\text{U}=-\frac{\text{GMm}}{\text{r}}$ KE of the satellite, $\text{K}=\frac{1}{2}\text{mv}^2=\frac{\text{GMm}}{2\text{r}}$$\bigg[\text{as v}=\sqrt{\frac{\text{GM}}{\text{r}}}\bigg]$ Total energy of the satellite, i.e., E = K + U $=\frac{\text{GMm}}{2\text{r}}-\frac{\text{GMm}}{\text{r}}=-\frac{\text{GMm}}{2\text{r}}$ For the sake of clarity, take $\frac{\text{GMm}}{2\text{r}}=\text{x}$ Clearly, U = -2x, K = x, E = -x The orbiting satellite loses energy due to viscous force acting on it due to atmosphere and as such it loses height. Let the new orbital radius be $\frac{\text{r}}{2}(\text{say})$ Clearly, U' = -4x K' = 2x E' = -2x Clearly, E' < E, U' < U and K' > K. Since, kinetic energy has increased, the velocity of the satellite increases.
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