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A mass m is placed at P a distance h alo...

A mass m is placed at P a distance h along the normal through the centre o of a thin circular ring of mass M and radius r (figure).

If the mass is moved further away such that OP becomes 2h by what factor the force of gravitation will decrease, if h = r?

Text Solution

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`implies` Gravitational force acting on an object of mass m, placed at point P at a distance h along the normal through the centre of a circular ring of mass M and radius r is given by
`F = (GMmh)/((r^(2) +h^(2))^(3//2) " "` (along `vecPO....(i)` )
When mass is displaced upto distance 2h, then
`F .= (GMmxx2h)/([r^2+(2h)^(2)]^(3//2))" "[ :. h.= 2h]`
`= (2GMmh)/((4^(2)+4h^(2))^(3//2))" " ....(ii)`
when h = r, in eq. (ii) ,
`F = (GMmxxr)/((r^2+r^2)^(3//2))impliesF =(GMm)/(2sqrt2r^2)`
and `F. = (2GMmr)/((r^2+4r^2)^(3//2))=(2GMm)/(5sqrt5r^2)`
`:. (F.)/(F) = (4sqrt2)/(5sqrt5)`
`implies F. = (4sqrt2)/(5sqrt5)F`
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