Question

The masses M in the two figures have identical values. Which of the two systems has a smaller gravitational potential energy? Why? M M- fig: (b) fig: (a)​

Answer 1

The second system has smaller gravitational potential energy.

Explanation:

The figure is attached.

We know that

Gravitational potential energy between two masses M and m, separated by a distance r is given by

$U=-\frac{GMm}{r}$

As shown, in figure (a), the distance between two masses M is

$\sqrt{r^2+r^2}=\sqrt{2r^2}=r\sqrt{2}$

Therefore the gravitational  potential energy is

$U=-\frac{GM^2}{r\sqrt{2}}$

In figure (b), the distance between the masses is

$2r$

Therefore the gravitational  potential energy is

$U'=-\frac{GM^2}{2r}$

Since $2r>r\sqrt{2}$

And the gravitational  potential energy has a negative sign

Therefore, in second case, the gravitational  potential energy will be smaller.

Hope this answer is helpful.

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