Question

If the distance between earth and sun is increased by 2 %, then find the percentage change in gravitational force acting between them.

Answer 1

Hint- Here, we will proceed by writing down the expression for gravitational force acting between earth and sun and then, also writing the same when the distance between earth and sun is increased by 2 %. Finally, we will divide these two expressions.

Formula Used- ${\text{F}} = \dfrac{{{\text{GMm}}}}{{{{\text{r}}^2}}}$.

Complete step-by-step solution -
Let the mass of the earth be m and the mass of the sun be M and they are initially separated by a distance of r.
As we know that the initial gravitational force F acting between any two bodies of masses M and m which are separated by a distance r is given by
${\text{F}} = \dfrac{{{\text{GMm}}}}{{{{\text{r}}^2}}}{\text{ }} \to {\text{(1)}}$ where G denotes the gravitational constant
Using the formula given in equation (1), the gravitational force F acting between the earth and sun is given by
${\text{F}} = \dfrac{{{\text{GMm}}}}{{{{\text{r}}^2}}}{\text{ }} \to {\text{(1)}}$
Now if the distance between earth and sun is increased by 2 %, the final distance between the earth and sun is given by
r’ = r + 2 % of r = r + 0.02r = 1.02r
Replacing r with r’ and F with F’ in the formula given by equation (1), we have
${\text{F'}} = \dfrac{{{\text{GMm}}}}{{{{\left( {{\text{r'}}} \right)}^2}}}$ where F’ denotes the final gravitational force acting between the sun and the earth separated by a distance of r’
By putting r’ = 1.02r in the above equation, we have
$ \Rightarrow {\text{F'}} = \dfrac{{{\text{GMm}}}}{{{{\left( {{\text{1}}{\text{.02r}}} \right)}^2}}} \\
   \Rightarrow {\text{F'}} = \dfrac{{{\text{GMm}}}}{{\left( {1.0404} \right){{\text{r}}^2}}}{\text{ }} \to {\text{(2)}} \\ $
Dividing equation (2) by equation (1), we get
$ \Rightarrow \dfrac{{{\text{F'}}}}{{\text{F}}} = \dfrac{{{\text{GMm}}}}{{\left( {1.0404} \right){{\text{r}}^2}}} \times \dfrac{{{{\text{r}}^2}}}{{{\text{GMm}}}} \\
   \Rightarrow \dfrac{{{\text{F'}}}}{{\text{F}}} = \dfrac{{\text{1}}}{{1.0404}}{\text{ }} \to {\text{(2)}} \\ $
Percentage change in quantity x = $\left( {\dfrac{{x - x'}}{x}} \right) \times 100 = \left( {1 - \dfrac{{x'}}{x}} \right) \times 100$
Using the above formula, we get
Percentage change in gravitational force = $\left( {1 - \dfrac{{{\text{F}}'}}{{\text{F}}}} \right) \times 100$
Using equation (2) in the above formula, we get
$ \Rightarrow $ Percentage change in gravitational force = $\left( {1 - \dfrac{{\text{1}}}{{1.0404}}} \right) \times 100 = \left( {\dfrac{{1.0404 - {\text{1}}}}{{1.0404}}} \right) \times 100 = \left( {\dfrac{{0.0404}}{{1.0404}}} \right) \times 100 = 3.88$ %
Therefore, the percentage change in gravitational force acting between them is 3.88 %.

Note- There are two types of percentage change in any quantity i.e., percentage increase in any quantity and percentage decrease in any quantity. Percentage increase in quantity x is given by $\dfrac{{{\text{Increase in x}}}}{{{\text{Original value of x}}}} \times 100$ % and percentage decrease in quantity x is given by $\dfrac{{{\text{Decrease in x}}}}{{{\text{Original value of x}}}} \times 100$ %.