Question

A trapeze artist of mass m swings on a rope of Length L. Initially, the trapeze artist is at rest and the rope makes an angle theta with the vertical. a) Find the tension in the rope when it is vertical. Why does your result for part a depend on L in the way it does? Use energy method only.

Answer 1

Theta is the maximum amplitude of the swinging motion, but they want the tension when the rope is vertical.

The tension at that time will equal the sum of the weight (mg) and the centripetal force required by the velocity and length (mV^2/L)

V will depend upon the amplitude of swinging, according to

(1/2)mV^2 = m g L(1 - cos theta)
V^2 = 2gL(1 - cos theta)

which comes from an energy conservation relationship. (1 - cos theta) is the vertical distance that the swing rises.

Now substitute

Maximum T = m (g + a)
= m (g + V^2/L)
= m [g + 2 g (1 - cos theta)]
= mg [1 + 2(1 - cos theta)]
= mg [3 - 2 cos theta]
There is no dependence upon L. It cancels out.

Answer 2

Thanks a lot for your immediate response