Case III: Vertical Circles  In vertical circular motion, the acceleration is not uniform as gravity speeds up objects while they fall and slows them down as they rise. Tension is greatest at the bottom of a vertical circle and approach minimum values while passing through the top of a vertical circle.



top


bottom


net F_{c} = T + mg



net F_{c} = T  mg


m(v^{2}/r) = T + mg



m(v^{2}/r) = T  mg


T = m(v^{2}/r)  mg



T = m(v^{2}/r) + mg

The following formula used to calculate the minimum, or critical, speed required for the block to pass through the top of a vertical circle is derived by taking the limit as T → 0 in the previous formula for centripetal force at the top of a vertical circle and solving for v:
m(v^{2}/r) = mg v^{2}/r = g v^{2} = rg v_{critical} = √(rg)
