Find the force of attraction on a particle of mass $m$ placed at the centre of a semicircular wire of length $L$ and mass $M$ .

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Solution

$L = π R$
$∴ R = \frac{L}{π}$
$F = \int_{0}^{π} d F sin θ$
$= \int_{0}^{π} \frac{G d M . m}{R^{2}} sin θ$
$= \int_{0}^{π} \frac{G (\frac{M}{π} d θ) m}{L^{2}} (π^{2}) sin θ$
$= \frac{2 π G m M}{L^{2}}$

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Find the force of attraction on a particle of mass m placed at the centre of a semicircular wire of length L and mass M.

L=πR∴R=LπF=∫π0dFsinθ=∫π0GdM.mR2sinθ=∫π0G(Mπdθ)mL2(π2)sinθ=2πGmML2

Find the force of attraction on a particle of mass $m$ placed at the centre of a semicircular wire of length $L$ and mass $M$ .

$L = π R$
$∴ R = \frac{L}{π}$
$F = \int_{0}^{π} d F sin θ$
$= \int_{0}^{π} \frac{G d M . m}{R^{2}} sin θ$
$= \int_{0}^{π} \frac{G (\frac{M}{π} d θ) m}{L^{2}} (π^{2}) sin θ$
$= \frac{2 π G m M}{L^{2}}$