Giant swing
Problem
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1) What is the minimum angular speed of the gymnast to make such a swing?
2) Find the force required by the arms to keep spinning.
3) Suppose that a gymnast makes one spin per second. Calculate his angular speed and the force on the arms at the bottom and the top positions.
State any assumptions that you make.
The picture is taken from http://www.sciencephoto.com/media/320527/view.
Getting Started
The average ratio for men $\frac{a}{b} = 0.56$ where $b$ is the height of a man $b = 1.82\mathrm{m}$ and $a$ is the position of mass centre measured from legs. The average length of arms is $32/33\;\mathrm{inches} = 82\mathrm{cm}$ and the average length from the shoulders to the top of the head is $30 \mathrm{cm}$. So, the centre of mass is about
1) We require that all kinetic energy at the bottom position will be transferred to the potential energy at the top position.
2) The force required by the arms can be found if we write the II-Newton's law for the man.
3) Suppose that the man is spinning with the constant angular speed then $T = 1\mathrm{s}$. Thus, the anglular speed is $\omega =\frac{2\pi}{T} = 2\pi\mathrm{s}^{-1} = 6.28 \mathrm{s}^{-1}$. Similarly as before we write the II-Newton's law at the bottom position to get that the tension in the arms.
At the top position