Calculation of Motion of a body in a vertical Circle: Let us consider, A body that has mass $m$ moving in a vertical circle of radius $r$. If at any instant the body is at position $P$ with angular displacement $\theta$ from the lower position $L$ of the circle. As shown in the figure below. The various forces acting on the body are: The weight $mg$ of the body, acting vertically downwards Tension $T$in the string acting along the $PO$. Now $mg$ can be resolved into two components: 1.) The horizontal component $mg cos\theta$ opposite to $T$ 2.) The vertical component $mg sin\theta$ act along tangent to the circle at $P$ So the net force on the body at position $P$ provides the necessary centripetal force required by the body $T-mg cos\theta = \frac{m v^{2}}{r}$ $T= \frac{m v^{2}}{r} + mg \: cos\theta \qquad(1)$ Tension and velocity at the highest position $H$ of the verticle circle: At the highest position $H$ the tension $T_{H}

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