Question

A large number of identical point masses m are placed alone x-axis at x= 0,1,2,4,..... The magnitude of gravitational force on mass at origin(x=0) will be
1. Gm^2
2. (4/3) Gm^2
3. (2/3) Gm^2
4. (5/4) Gm^2

Answer 1

Answer on Question 48973, Physics, Astronomy — Astrophysics

A large number of identical point masses m are placed alone x-axis at x= 0,1,2,4,….. The magnitude of gravitational force on mass at origin(x=0) will be 1. Gm^{2} 2. (4/3) Gm^{2} 3. (2/3) Gm^{2} 4. (5/4) Gm^{2}

Solution

Gravitational force from single mass is

$F=\frac{Gm^{2}}{r^{2}}$

where r is distance to origin. Hence, we have to sum all the masses, and the force will be

$F_{total}=Gm^{2}\sum_{1}^{\infty}\frac{1}{(2^{n})^{2}}=Gm^{2}\sum_{0}^{\infty}\frac{1}{2^{n+1}}=Gm^{2}\sum_{1}^{\infty}\frac{1}{2^{n}}=Gm^{2}$

Answer is 1. Gm^{2}

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