Question

Infinite number of masses each $1kg$ are placed along the $X–axisatx=\pm lm,\pm 2m,\pm 4m,\pm 8m,\pm 16m...$. The magnitude of the resultant gravitational potential in terms of gravitational constant$G$ at the origin $(x=0)$ is

• A. $\frac{G}{2}$
• B. $G$
• C. $2G$
• D. $4G$
• E. $8G$

Answer 1

The correct option is D

$4G$

Solution:

Step 1: Given information

$X–axisatx=\pm lm,\pm 2m,\pm 4m,\pm 8m,\pm 16m...$

Masses $=$ $1kg$ (At values of x on the X-axis)

Step 2: Calculate the magnitude of the resultant gravitational potential in terms of the gravitational constant$G$ at the origin $(x=0)$

Know that at a distance $r$ the gravitational potential of two masses $m_1$ and $m_2$ is given by

$g=\frac{G{m}_1{m}_2}{r}$

Where,

$G=$Gravitational constant

$m_1=$Mass of object A

$m_2=$Mass of object B

$r=$Distance between object A and object B

Therefore, the gravitational potential for the given masses at the origin may be calculated using this method,

$$\begin{gathered} u=2\left[-\frac{G\times 1}{1}-\frac{G\times 1}{2}-\frac{G\times 1}{4}....\right] \\ u=-2G\left[1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}...\right] \\ u=-2G\left[\frac{1}{1-{\displaystyle \frac{1}{2}}}\right] \\ u=-4G \end{gathered}$$

Hence, the correct option is (D).