Question

Review: Why is the following situation impossible? An athlete tests her hand strength by having an assistant hang weights from her belt as she hangs onto a horizontal bar with her hands. When the weights hanging on her belt have increased to 80% of her body weight, her hands can no longer support her and she drops to the floor. Frustrated at not meeting her hand-strength goal, she decides to swing on a trapeze. The trapeze consists of a bar suspended by two parallel ropes, each of length, $\frac{\mathbf{4}\mathbf{mg}}{\mathbf{k}}$, allowing performers to swing in a vertical circular arc (Fig. P8.70). the athlete holds the bar and steps off an elevated platform, starting from rest with the ropes at an angle ${\mathbf{\theta }}_{\mathbf{i}}\mathbf{=}{\mathbf{60}}^{\mathbf{o}}$with respect to the vertical. As she swings several times back and forth in a circular arc, she forgets her frustration related to the hand-strength test. Assume the size of the performer’s body is small compared to the I length, and air resistance is negligible.

Short answer

The tension in the hand at the lowest point is more than they can withstand. Therefore, the given condition is impossible.

Step-by-step solution

Step 1:

The mathematical representation of the motion of an object, considering all the forces acting on it, is known as the equation of motion of that object. Free body diagrams are often used to derive the equation of motion of any object that is part of a system, represented in the diagram. Generally, the equation of motion is of the form $\sum \mathbf{F}\mathbf{=}\mathbf{ma}.$

Here:

$\mathrm{F}=$magnitude of the force exerted on the body.

$\mathrm{a}=$magnitude of the acceleration in the body.

Given Data

Weight of the body$\mathrm{W}=\mathrm{mg}.$

The angle that the rope makes with the vertical at the starting point-$\mathrm{\theta }=60°.$

The length of the ropes $\mathrm{l}=\frac{4\mathrm{mg}}{\mathrm{k}}$

The maximum tension that the performer’s hand can bear-${\mathrm{T}}_{\max }=1.8\mathrm{mg}.$

Step 3:

If $\overrightarrow{\mathrm{F}}$is the tension in the ropes while swinging, then from the given diagram, we get:

$\begin{array}{c}\mathrm{F}-\mathrm{mgcos\theta }=\mathrm{m}\frac{{\mathrm{v}}^2}{\mathrm{l}}\\ \mathrm{F}=\mathrm{mgcos\theta }+\mathrm{m}\frac{{\mathrm{v}}^2}{\mathrm{l}}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(1\right)\end{array}$

Here, $\mathrm{m}\frac{{\mathrm{v}}^2}{\mathrm{l}}$is the centripetal force in the ropes and$\mathrm{v}$is the velocity with which the performer oscillates.

At the bottom of the swing,$\mathrm{\theta }=0^{\mathrm{o}}$, so

$\mathrm{F}=\mathrm{mg}+\mathrm{m}\frac{{\mathrm{v}}^2}{\mathrm{l}}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(2\right)$

The maximum tension that the performer’s hand can withstand is $1.80mg$. By applying the law of conservation of mechanical energy on the performer-earth system, we get:

$\begin{array}{c}\mathrm{mgl}(1-\mathrm{cos\theta })=\frac{1}{2}{\mathrm{mv}}^2\\ \mathrm{mgl}(1-\cos {60.0}^{\mathrm{o}})=\frac{1}{2}{\mathrm{mv}}^2\\ \frac{{\mathrm{mv}}^2}{\mathrm{l}}=2\mathrm{mg}\left(1-\frac{1}{2}\right)\\ \frac{{\mathrm{mv}}^2}{\mathrm{l}}=\mathrm{mg}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left(3\right)\end{array}$

From equation (2) and (3):

$\begin{array}{c}\mathrm{F}=\mathrm{mg}+\mathrm{m}\frac{{\mathrm{v}}^2}{\mathrm{l}}\\ =\mathrm{mg}+\mathrm{mg}\\ =2\mathrm{mg}.\end{array}$

This is the force acting on the performer’s hand at the point of the least height. As the tension in the performer’s hand at this point is greater than the tension it can withstand, she will fall down. So, the given situation is impossible.