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

Answer 1

Answer: (C)

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

Answer 2

If two particles of mass $m$ are placed $x$ distance apart then force of attraction $\frac { G m m } { x ^ { 2 } } = F$ (Let)

Now according to problem particle of mass $m$ is placed at the centre (P) of square. Then it will experience four forces

$F _ { P A } =$ force at point P due to particle $A = \frac { G m m } { x ^ { 2 } } = F$

Similarly $F _ { P B } = \frac { G 2 m m } { x ^ { 2 } } = 2 F$ , $F _ { P C } = \frac { G 3 m m } { x ^ { 2 } } = 3 F$ and

$F _ { P D } = \frac { G 4 m m } { x ^ { 2 } } = 4 F$

Hence the net force on P

$\therefore$ $= 2 \sqrt { 2 } \frac { G m ^ { 2 } } { ( a / \sqrt { 2 } ) ^ { 2 } }$ [$x = \frac { a } { \sqrt { 2 } } =$ half of

the diagonal of the square]

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