Chapter 11, Problem 27P

To determine

To Find: The spring stiffness constant and unstretched length.

Answer to Problem 27P

The spring stiffness constant $113.73\text{ N/m}$ and unstretched length of the bungee cord is $19.4\text{ m}$ .

Explanation of Solution

Given info:

Mass, $m=65\text{ kg}$

Total length, $l=25\text{ m}$

Time, $t=38\text{ s}$

Number of cycles, $n=8$

Formula used:

Write the expression for the time period.

  $T=\frac{\text{time}}{\text{no of cycles}}$ ........... Equation(1)

Write the expression of time period in terms of mass and stiffness.

  $T=2\pi \times \sqrt{\frac{m}{k}}$ ........... Equation(2)

Write the expression showing relation between the spring force and gravitation force.

  $k\times \Delta x=m\times g$ ........... Equation(3)

Calculation:

Substitute the given values in equation (1),

  $T=\frac{38}{8}=4.75\text{ s/cycle}$

Calculate value of stiffness using equation (2).

  $\begin{array}{l}{T}^2=4{\pi }^2\times \frac{m}{k}\\ k=\frac{4\times {\pi }^2\times m}{T^2}=113.73\text{ N/m}\end{array}$

Stretched of a bungee cord needs to provide a force equal to weight of jumper when it is at equilibrium position.

  $\Delta x=\frac{m\times g}{k}=\frac{65\times 9.8}{113.73}=5.6\text{ m}$

Unstretched length of bungee end

  $\begin{array}{c}\text{= Total length - Stretched Length}\\ \text{= }25-5.6\\ =19.4\text{ m}\end{array}$

Conclusion:

Thus, the spring stiffness constant $113.73\text{ N/m}$ and unstretched length of the bungee cord is $19.4\text{ m}$ .