Question

Infinite number of point masses each equal to m
are placed at x =1, x = 2, x = 4, x = 8......, what
is the total gravitational potential at x = 0?
(1) - Gm
(2) - 2Gm
(3) - 4Gm
(4) - 8Gm
​

Answer 1

Answer:

The total gravitational potential is -2Gm.

(2) is correct option.

Explanation:

Given that,

Infinite number of point masses each equal to m  are placed at x =1, x = 2, x = 4, x = 8......,

The resultant potential will be the potential from each one added to one another

We need to calculate the total gravitational potential at x = 0

Using formula of gravitational potential

$U=-\dfrac{Gm}{r^2}$

Put the value into the formula

$U=-(\dfrac{Gm}{1}+\dfrac{Gm}{2}+\dfrac{Gm}{4}+\dfrac{Gm}{8}.....)$

$U=-Gm(1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}......)$

Here, $1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}....$ is a G.P series

So, $T_{n}=\dfrac{a}{1-r}$

$T_{n}=\dfrac{1}{1-\dfrac{1}{2}}$

$T_{n}=2$

Where, a = first term

r = common ratio

$U=-2GM$

Hence, The total gravitational potential is -2Gm.