Question

Three point masses each of mass ${}^{\prime }{m}^{\prime }$ are kept at three vertices of a square of side ${}^{\prime }{a}^{\prime }$ as shown in figure. Find gravitational field strength at point $\text{O}$.

• A. $\sqrt{2}\frac{Gm}{a^2}$, towards $\text{B}$
• B. $(\sqrt{2}+\frac{1}{2})\frac{Gm}{a^2}$, towards $\text{O}$
• C. $(\sqrt{2}+\frac{1}{2})\frac{Gm}{a^2}$, towards $\text{B}$
• D. None of the above

Answer 1

The correct option is C $(\sqrt{2}+\frac{1}{2})\frac{Gm}{a^2}$, towards $\text{B}$

Gravitational field at distance $r$ due to a point mass $m$, $$E=\frac{Gm}{r^2}$$Let, the gravitational fields at point $\text{O}$ due to the masses placed at vertices $\text{A, B}$ and $\text{C}$ be $E_A,E_B$ and $E_C$ respectively.

From the figure,
$E_A=E_C=\frac{Gm}{a^2}=E\left(\text{say}\right)$ (towards the point masses as shown in figure)

And
$E_B=\frac{Gm}{(\sqrt{2}a{)}^2}=\frac{Gm}{2a^2}=\frac{E}{2}$

Resultant of $E_A$ and $E_C$ is

$E_{AC}=\sqrt{E_A^2+E_C^2}=\sqrt{E^2+E^2}=\sqrt{2}E$ (towards $\text{B}$).

So, the net gravitational field strength at point $\text{O}$ is

$E_{net}=E_{AC}+E_B$

$\Rightarrow E_{net}=\sqrt{2}E+\frac{E}{2}$

$\Rightarrow E_{net}=(\sqrt{2}+\frac{1}{2})E$

$\therefore E_{net}=(\sqrt{2}+\frac{1}{2})\frac{Gm}{a^2}$ (towards $B$)

Hence, option (c) is the correct answer.

why this question: This question will help the students to understand the application of the superposition principle.