A semicircular wire has a length L and mass M A particle of mass m is placed at the centre of the circle.
Find the gravitational attraction on the particle due to the wire.

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Solution

In the semicircle, we can consider, a small element $d θ$
Then, mass of R $d θ$ $= (\frac{M}{L}) R d θ d F = \frac{G M R d θ m}{L R^{2}} d F_{1} = 2 d F S i n c e = \frac{2 G M m}{L R} sin θ d θ ∴ F = \int_{0}^{\frac{π}{2}} \frac{- 2 G M m}{L R} sin θ d θ = \frac{2 G M m}{L R} [- C o s θ]_{0}^{\frac{π}{2}} = - 2 \frac{G M m}{L R} (- 1) = \frac{2 G M m}{L R} = \frac{2 G M m}{\frac{L L}{π}} = \frac{2 π G M m}{L^{2}}$

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A semicircular wire has a length L and mass M A particle of mass m is placed at the centre of the circle. Find the gravitational attraction on the particle due to the wire.

In the semicircle, we can consider, a small element dθ Then, mass of R dθ =(ML)RdθdF=GMRdθmLR2dF1=2dFSince=2GMmLRsinθdθ∴F=∫π20−2GMmLRsinθdθ=2GMmLR[−Cosθ]π20=−2GMmLR(−1)=2GMmLR=2GMmLLπ=2πGMmL2

A semicircular wire has a length L and mass M A particle of mass m is placed at the centre of the circle.
Find the gravitational attraction on the particle due to the wire.