Gravitation Question 350
Question: The radius of the earth is reduced by 4%. The mass of the earth remains unchanged. What will be the change in escape velocity?
Options:
A) Increased by 2%
B) Decreased by 4%
C) Increased by 6%
D) Decreased by 8%
Show Answer
Answer:
Correct Answer: A
Solution:
- Escape velocity $ [=v=\sqrt{\frac{2GM}{R}}] $
$ [\Rightarrow v^{2}=\frac{2GM}{R}] $ ….(i)
$ [v^{2}=\left( 2GM \right){{R}^{-1}}] $
Differentiating both sides, we get,
$ [2v\frac{dv}{dR}=-\frac{2GM}{R^{2}}\Rightarrow v\frac{dv}{dR}=\frac{GM}{R^{2}}] $ ….(ii) Dividing (ii) by (i),
$ [\frac{1}{v}\frac{dv}{dR}=-\frac{1}{2R}] $
$ [\Rightarrow \left| \frac{dv}{v} \right|\times 100=\frac{1}{2}\times 4=2]$%
$ [\therefore ]$ If the radius decreases by 4%, escape velocity will increase by 2%.
