Question

A ring of mass m1 and radius r is fixed in space at some location an external agent brings a point mass m2 from infinity to centre of ring work done by external agent will be

Answer 1

All point on the circumference of the ring are at distance R 2 +x 2

from the center O of the ring.

Force on the mass m at P : F= R 2+x 2

GMm ×(2cosθ) where cosθ= R 2 +x 2

x

Due to symmetry, vertical components of the forces from two symmetrical elements cancel out, the horizontal components add up, hence we get a factor of 2.

∴F= R 2 +x 2

GMm ×2 R 2 +x 2

x

When F is maximum,

dxdF=0

⇒(R 2 +x 2 ) − 23 +x(− 23 (R 2 +x 2 ) − 23 −1

×2x=0

⇒(R 2 +x 2 )=3x 2

⇒x= 2R..

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$All point on the circumference of the ring are at distance R2+x2from the center O of the ring.</p><p></p><p>Force on the mass m at P : F=R2+x2GMm×(2cosθ) where cosθ=R2+x2x</p><p>Due to symmetry, vertical components of the forces from two symmetrical elements cancel out, the horizontal components add up, hence we get a factor of 2.</p><p>∴F=R2+x2GMm×2R2+x2x</p><p>When F is maximum, dxdF=0</p><p>⇒(R2+x2)−23+x(−23(R2+x2)−23−1×2x=0</p><p>⇒(R2+x2)=3x2</p><p>⇒x=2R</p><p></p><p></p><p>$

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