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Question

A large number of identical point masses each of mass m are placed along x-axis, at x=0,1,2,4,... The magnitude of gravitational force on the mass at origin (x=0), will be



A
Gm2
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B
43Gm2
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C
23Gm2
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D
54Gm2
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Solution

The correct option is B 43Gm2
Let, F1, F2, F4, F8,.... be the forces of gravitation on mass m placed at x=0 due masses placed at x=1, 2, 4, 8... respectively.


The gravitational force between any two point masses is given by F=Gm1m2r2 According to principle of superposition, the net gravitational force on mass m placed at x=0 can be written as

F=F1+F2+F4+F8...

F=Gm212+Gm222+Gm242+Gm282+...

F=Gm2(11+14+116+164...)...(1)

The above expression consists a infinite G.P with common ratio, r=14

So, sum of the G.P, S=a1r

where a is the first term r is the common ratio.

S=1114=43

From equation (1),

F=F1+F2+F4+F8......=Gm2S

F=43Gm2

Hence, option (b) is the correct answer.
Why this question:

To make students understand that gravitational force follows the principle of superposition " The net force on a particle is equal to the vector sum of all the forces acting on that particle".

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