Question

Four particles of masses m, 2m, 3m and 4m are kept in sequence at the corners of a square of side a. The magnitude of gravitational force acting on a particle of mass m placed at the centre of the square will be

• A. $\frac{24m^{2}G}{a^2}$
• B. $\frac{6m^{2}G}{a^2}$
• C. $\frac{4\sqrt{2}G{m}^2}{a^2}$
• D. $Zero$

Answer 1

The correct option is C

$\frac{4\sqrt{2}G{m}^2}{a^2}$

If two particles of mass m are placed x distance apart then force of attraction $\frac{Gmm}{x^2}=F$ (Let) Now according to problem particle of mass m is placed at the centre (P) of square. Then it will experience four forces

$F_{PA}$ = force at point P due to particle $A=\frac{Gmm}{x^2}=F$

Similarly $=F_{PB}=\frac{G2mm}{x^2}=2F,F_{PC}=\frac{G3mm}{x^2}=3Fand{F}_{PD}=\frac{G4mm}{x^2}=4F$

Hence the net force on ${\overrightarrow{F}}_{net}=2\sqrt{2}\frac{Gmm}{x^2}=2\sqrt{2}\frac{G{m}^2}{{\left(\frac{a}{\sqrt{2}}\right)}^2}$

[$x=\frac{a}{\sqrt{2}}$ = half of the diagonal of the square ]

$=\frac{4\sqrt{2}G{m}^2}{a^2}$