Question

If the radius of earth increase by 1% and its mass remains same then acceleration due to gravity on the surface of earth will-
A. Increase by 2%
B. Decrease by 2%
C. Increase by 1%
D. Decrease by 1%

Answer 1

Hint: Write the equation for the gravitational force acting on a body on the surface of the Earth. From this equation derive the formula for acceleration due gravity. Now replace the value of radius with the new radius value and find out the new value for acceleration due to gravity. Compare both the values.

Complete step by step solution:
The gravitational forces acting on a body on the surface of the Earth is given by
$F={\displaystyle \frac{GMm}{R^2}}$
Where,
G is universal constant of gravitation
M is the mass of the Earth
m is the mass of the body
Equating this Equation with $F=ma$, where a is the acceleration of a body, we get
$\begin{array}{rl}& ma={\displaystyle \frac{GMm}{R^2}}\\ & \therefore a={\displaystyle \frac{GM}{R^2}}\end{array}$
This is the acceleration with a body falling towards the center of the Earth.
We call this acceleration as acceleration due to gravity and denote it by ‘g’.
Thus,
$g={\displaystyle \frac{GM}{R^2}}$
Now, this is the value of acceleration of a body at the surface of the Earth.
Given that, radius of earth is increased by 1% and its mass mass remains same, then the new radius is
$$\begin{array}{rl}& R^{\prime }=R+{\displaystyle \frac{1}{100}}R\\ & R^{\prime }=1.01R\end{array}$$
The value acceleration due to gravity becomes
$g^{\prime }={\displaystyle \frac{GM}{{(R^{\prime })}^2}}$
$\begin{array}{rl}& g^{\prime }={\displaystyle \frac{GM}{{(1.01R)}^2}}\\ & \therefore g^{\prime }=0.98g\end{array}$
Since, $g^{\prime }<g$, the value of acceleration due to gravity decreases.
$\begin{array}{rl}& \mathrm{\%}\text{ decrease in g = }{\displaystyle \frac{g-g^{\prime }}{g}}\times 100\\ & \mathrm{\%}\text{ decrease in g}={\displaystyle \frac{g-0.98g}{g}}\times 100\\ & \mathrm{\%}\text{ decrease in g}=2\end{array}$
Thus, the acceleration due to gravity on the surface of the Earth will decrease by 2%
Answer- B. Decrease by 2%

Note: Acceleration due to gravity is the acceleration of the body with which it falls towards the center of the Earth due to the gravitational force of attraction. It depends on the mass of the Earth and the distance of the body from the center of the Earth.