Question

1) A child is swinging from a suspended rope 4.0 m long that will break if the tension equals twice the weight of the child.

a) What is the greatest angle the rope can make with the vertical during a swing if the rope is not to break?

b) If the angle slightly exceeds that found in part (a), what is the speed of the child when the rope breaks?

Answer 1

Let the greatest angle made by the rope with vertical to be . Now for rope not to break we have

T < 2mg

at the angle , the child will be at momentarily rest but it will acquire a potential energy which is equal to

mg4cos = mv²/2

=> v² = 8gcos

Now the centrifugal force will be mv²/l = m8gcos/4

Now this is balanced by T = 2mgcos + mg

For max ; 2mg = 2mgcos + mg

=> mg = 2mgcos

=> cos = 1/2

=> = 60^o

If the angle slightly increases then the velocity at bottom where the rope will break will be v = = 2 = 6.26 m/s