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

= m(u^2/r - 2g + 2g cos theta + g cos theta)

= m(u^2/r - 2g + 3g cos theta)

T = m(u^2/r + 3g cos theta - 2g)

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4 Tension at Key Points

At the bottom (theta = 0): T_bottom = m(u^2/r + g)

At the top (theta = pi): T_top = m(u^2/r - 5g) (since cos pi = -1, and v_top^2 = u^2 - 4gr)

Also from the radial equation at the top: T_top + mg = mv_top^2/r, so T_top = mv_top^2/r - mg.

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5 Condition for Complete Circles (String)

For a particle on a string, the string must remain taut. The critical point is the top, where tension is smallest.

T_top >= 0

mv_top^2/r - mg >= 0

v_top^2 >= gr

Using energy: v_top^2 = u^2 - 4gr. So:

u^2 >= 5gr (condition for complete circles on a string)

At the critical case (u^2 = 5gr): T_top = 0, v_top = sqrt(gr).

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6 Worked Example 1

A particle on a string of length 0.5 m moves in a vertical circle. Find the minimum speed at the bottom for complete circles.

u^2 >= 5gr = 5 * 9.8 * 0.5 = 24.5

u >= sqrt(24.5) = 4.95 m s^(-1)

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7 Worked Example 2

A particle on a string of length 2 m passes through the bottom at 10 m s^(-1). Find the tension at the top.