Question

Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. 5–42). If his arms are capable of exerting a force of 1150 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 78 kg and the vine is 4.7 m long.

FIGURE 5-42. Problem 18

Short answer

The maximum speed that Tarzan can tolerate at the lowest point of his swing is 4.82 m/s.

Step-by-step solution

Step 1. Understanding the centripetal force acts on the Tarzan

When the Tarzan of mass m moves in a circular arc of radius rhaving a constant speed v, it will need a force that will keep the Tarzan moving in the circular motion. This type of force acts on the Tarzan towards the arc center, which can be termed the centripetal force.

Mathematically, it is written as:

$F_{\mathrm{R}}=\frac{m{v}^2}{r}$

Here, Tarzan is moving in a circle with the centripetal force pointing upwards at the lowest point of his swing.

Step 2. Identification of the given information

• The maximum tension that can be created in the vine is, $T_{\max .}=1150\mathrm{N}$.
• The length of the vine is, l = 4.7 m.
• The mass of the Tarzan is, m = 78 kg.

Let the maximum speed that Tarzan can tolerate at the lowest point of the swing be v.

Step 3. Representation of the free body diagram of the Tarzan

The free-body diagram of the Tarzan at the lowest point of the swing is:

Here, the forces acting on the Tarzan are. his weight (mg) acting downwards and tension in the vine acting upwards (T).

Step 4. Determination of the expression for speed

As it is clear from the figure, the net force acting on the Tarzan in the vertical direction is, $\left(T-mg\right)$. This force must provide the centripetal force required by the Tarzan to swing in an arc. Thus, the centripetal force is:

$\begin{array}{c}{F}_R=T-mg\\ m\frac{v^2}{r}=T-mg\end{array}$

Thus, the expression for the speed of the Tarzan can be written as:

$v=\sqrt{\frac{\left(T-mg\right)r}{m}}$

Step 5. Determination of the maximum speed that Tarzan can tolerate

At the lowest point of the swing, the speed will be the maximum if tension in the vine is maximum. Thus, the maximum speed is:

$v_{\max }=\sqrt{\frac{\left(T_{\max }-mg\right)r}{m}}$

Substitute all the values in the above equation.

$\begin{array}{c}{v}_{\max }=\sqrt{\frac{\left[1150\mathrm{N}-\left(78\mathrm{kg}\times 9.8\mathrm{m}/{\mathrm{s}}^2\right)\left(\frac{1\mathrm{N}}{1\mathrm{kg}·\mathrm{m}/{\mathrm{s}}^2}\right)\right]\times 4.7\mathrm{m}}{78\mathrm{kg}}}\\ =\sqrt{23.235}{\mathrm{m}}^2/{\mathrm{s}}^2\\ =4.82\mathrm{m}/\mathrm{s}\end{array}$

Thus, the maximum speed that Tarzan can tolerate is 4.82 m/s.