Question

derive the formula center of mass of a semi circular ring please help i found in many books but the proof is not given but in our the derivation will come for 4 marks

Answer 1

Total length of semi circular wire = πR

Elemental length = Rdθ

$$\mathrm{dm} = \frac{\mathrm{mR} \mathrm{d}\theta}{\pi \mathrm{R}} = \frac{\mathrm{m} \mathrm{d}\theta}{\pi}$$ $$\begin{array}{l} S_{0}, y = \int_{0}^{\pi} \frac{1}{m} \, y \, dm \\ = \frac{1}{m} \int_{0}^{\pi} R \sin \theta \frac{m}{\pi} \, d\theta \\ = \frac{2R}{\pi} \end{array}$$