Vertical Circular Motion

Vertical circular motion is more complex than horizontal because the speed of the particle varies as it moves around the circle -- gravity does work, speeding the particle up on the way down and slowing it on the way up. The critical question is: what conditions are needed for complete circles?

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1 Forces at a General Point

Consider a particle of mass m on a string of length r, moving in a vertical circle. At an angle theta from the lowest point:

Towards centre (radially): T - mg cos theta = mv^2/r

Along the tangent: -mg sin theta = m(dv/dt) (this changes the speed)

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2 Energy Equation

Using conservation of energy (if no friction/air resistance):

At the bottom (speed u): KE = (1/2)mu^2, GPE = 0.

At angle theta (speed v, height h = r - r cos theta = r(1 - cos theta)):

(1/2)mu^2 = (1/2)mv^2 + mgr(1 - cos theta)

v^2 = u^2 - 2gr(1 - cos theta)

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3 Tension at a General Point

From the radial equation:

T = mv^2/r + mg cos theta = m(u^2 - 2gr(1 - cos theta))/r + mg cos theta