Chapter 7, Problem 2RAT

When the distance between two stars is reduced by 1/2 , the force between them

(a) decreases by 1/2.

(b) decreases by 1/4.

(c) increases by twice times as much.

(d) increases four times as much.

To determine

The correct option for the statement "When the distance between two stars is reduced by 1/2, the force between them".

Answer to Problem 2RAT

The correct option for the statement "When the distance between two stars is reduced by 1/2, the force between them" is option (d).

Explanation of Solution

The force acting between the two stars is gravitational force. The gravitational force between two masses has an inverse square relation with the distance between them.

Write the relation between the force and the distance.

$F\propto \frac{1}{d^2}\mathrm{.........}\left(\text{i}\right)$

Here,

F is the force

d is the distance between the stars

When r becomes half, the proportionality relation between the new force acting between the two stars and the distance becomes,

$\begin{array}{l}f\alpha \frac{1}{{\left(\frac{d}{2}\right)}^2}\\ f\alpha \text{}\frac{4}{d^2}\mathrm{.........}\left(\text{ii}\right)\end{array}$

Here,

f is the new force

Divide equation (ii) by equation (i).

$\begin{array}{l}\frac{f}{F}=\frac{\frac{4}{d^2}}{\frac{1}{d^2}}\\ f=4F\end{array}$

Thus, when the distance between two stars is reduced by 1/2, the force between them increases four times as much.

Conclusion:

Therefore, the correct option for the statement "When the distance between two stars is reduced by 1/2, the force between them" is option (d).