A father wants to give his young son a thrill by swinging him all the way around

Question

A father wants to give his young son a thrill by swinging him all the way around the support bar of a swing without the chains going slack. The chains are of 3 m in length each, and the boy weighs 2-15 N. The father knows that some energy will be dissipated through friction, so he plans for the swing to start with an initial speed faster (by 2 m/s) than required to just make it around in the absence of friction. If he releases the boy 1 m above the lowest point of the swing, how fast must the boy be going at release?

Explanation / Answer

At the highest point, height of kid from the point of releae = 6 - 1 = 5m = h
potrential energy = mgh ( where m is mass of kid)
for the chain not to go slack, gravity must be balanced by centripital force
so if velocity of kid at maximum height is v
mv^2/r = mg [ where r is radius of the swing]
v^2 = rg
KE of the kid = 0.5mv^2 = 0.5mrg
Net energy at the top = 0.5mrg + mgh = Net energy at the botttom ( ignoring friciton)
KE at bottom = 0.5mu^2 [ u si initial velocity of kid]
so, 0.5mu^2 = 0.5mrg + mgh
u^2 = rg + 2gh = 3g + 2g*5 = 13g
u = 11.29 m/s

accounting for speed
initial speed = 11.29 + 2 = 13.29 m/s

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