Giant Swing
Calculate the forces on the joints of a spining gymnast
Problem
A giant swing is an artistic gymnastics element in which a gymnast rotates around an axis while in a fully extended position, as in this image.
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.
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 $$h = 1.82\mathrm{m} - 0.56\times 1.82\mathrm{m} + 0.82\mathrm{m} - 0.3\mathrm{m} = 1.32\mathrm{m}$$ measured from the wrist.
1) We require that all kinetic energy at the bottom position will be transferred to the potential energy at the top position.
$$\frac{mv^2}{2} = 2mgh$$ where $m$ is the mass of the man, $g = 9.81\mathrm{m/s}^2$ and $v$ is the speed of the centre of mass. Thus, $v = 2\sqrt{gh}$. The angular speed $$\omega = \frac{v}{h} =\frac{2\sqrt{gh}}{h} = 2\sqrt{\frac{g}{h}} = 2\sqrt{\frac{9.81\mathrm{m/s}^2}{1.32\mathrm{m}}} = 5.45 \mathrm{s}^{-1}\;.$$ Notice that the answer is independent of the mass of the man.
2) The force required by the arms can be found if we write the II-Newton's law for the man.
$$ma = F_{T} - mg$$ where $a$ is the centripetal acceleration, $m$ the mass of the man and $F_{T}$ is the tension in the arms. So, $$F_{T} = m(a + g) = m\left(\frac{v^2}{h} + g\right) = m\left(\omega^2h + g\right)\;.$$ Use the expression for $\omega$ from the previous part to get $F_{T} = 5mg$. Suppose that an average man weights $80\mathrm{kg}$. $$F_{T} = 5\times80\mathrm{kg}\times9.81\mathrm{m/s}^2 = 3.9\mathrm{kN}$$ It is a huge force. Notice that this time the answer is independent of the position of the centre of mass.
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. $$F_{B} = m(\omega^2h + g) = 80\left(4\pi^2\mathrm{s}^{-2}\times1.32\mathrm{m} + 9.81\mathrm{m/s}^2\right) = 4.95\mathrm{kN}$$ or it is about $F_{B} = 6.3mg$.
At the top position $$F_{T} = m(\omega^2h - g) = 80\mathrm{kg}\left(4\pi^2\mathrm{s}^{-2}\times1.32\mathrm{m} - 9.81\mathrm{m/s}^2\right) = 4.3mg = 3.38\mathrm{kN}\;.$$ This time the answer is dependent on both the position centre of mass and the mass of the man.