Gravitation-Where am I stuck?

Classical Mechanics Level 2

Four particles of masses $m$ , $2m$ , $3m$ and $4m$ are placed in a clockwise manner at $4$ corners of a square of side $a$ . Find the gravitational force acting on a particle of mass $m$ placed at the center.

If the answer is in the form $\frac { m\sqrt { n } G{ M }^{ 2 } }{ { a }^{ 2 } }$ , find $m+n$ .

The answer is 6.

1 solution

Department 8
Sep 11, 2015

I don't have pictures so the solution may not be easy to interpret.

Note that mass $m$ at center has a distance of $\frac{\sqrt{2}a}{2}$ . By Geometry we see the corner masses $m, \quad 3m$ are opposite to each other and same as with $2m, \quad 4m$ . By Newton's Second Law we see mass $4m$ and $3m$ pull the center mass to each other by an equal force of $\frac{4GM^{2}}{a^{2}}$ . The magnitude of the resultant force can be calculated by vector summing which would give $\frac{4\sqrt{2}GM^{2}}{a^{2}}$ .

Gravitation-Where am I stuck?

The answer is 6.

1 solution

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