Question

Infinite number of masses, each of 1 kg, are placed along the x-axis at x = +- 1m, +- 2m, +-4m, +- 8m, +- 16m.. The gravitational of the resultant gravitational potential in term of gravitaitonal constant G at the origin (x = 0) is

Answer 1

Given :

Mass of each body = 1 kg

Masses are placed along X-axis at x = +- 1m , +- 2m , +-4m, +-8m ,+-16m ......

To Find :

The resultant of gravitational potential of all the masses in terms of gravitational constant G at the origin = ?

Solution :

We know that the gravitational potential of two masses $m_1$ and $m_2$  at a distance r is given as :

$= \frac{Gm_1 m_2}{r}$

Here , G is gravitational constant

So, according to this formula the gravitational potential for the given masses   at origin will be given as :

= $2\times (\frac{G\times 1 \times 1}{1} + \frac{G\times 1 \times 1}{2}+\frac{G\times 1 \times 1}{4} +......+ \frac{G\times 1 \times 1}{\infty})$

$= 2 \times (\frac{G}{1}+\frac{G}{2}+ \frac{G}{4}+........)$

Here it is forming a series of infinite GP, Since the formula for sum of infinite GP is given as = $\frac {a}{1-r}$

where a is 1st term and r is the common ratio .

So, here sum of the GP is :

$=\frac{(\frac{G}{1})}{(\frac{1}{2}) }$

= 2G

So , total gravitational potential at origin is $2\times 2G$   = 4G

Answer 2

Trick

Do as above and use this tip

As the body are placed on both sides of the x-axis , we should multiply by 2

to get the answer