A bungee jumper, of mass 70 kg, ties one end of an elastic rope of length 65 m and stiffness 1

step 1 So to start off this problem, M equals 160 over 386 .4, and that is in pounds per second squared

A bungee jumper, of mass 70 kg, ties one end of an elastic...

A bungee jumper, of mass $70 \mathrm{~kg},$ ties one end of an elastic rope of length $65 \mathrm{~m}$ and stiffness $1.75 \mathrm{kN} / \mathrm{m}$ to a bridge and the other end to himself and jumps from the bridge (Fig. 2.68 ). Assuming the bridge to be rigid, determine the vibratory motion of the jumper about his static equilibrium position.

Key Concepts

Natural Frequency

The natural frequency of a mass-spring system is a measure of how fast the system oscillates when disturbed from equilibrium, and it is derived from the system’s parameters such as the mass and the stiffness of the spring. It is calculated by taking the square root of the ratio of the spring constant to the mass. This parameter is crucial because it sets the time scale of the vibratory motion and is used to analyze the dynamic response of the system.

Simple Harmonic Motion (SHM)

Once the mass is displaced from its static equilibrium, if the restoring force follows Hooke’s Law, the motion can often be approximated as simple harmonic motion. SHM is characterized by sinusoidal oscillations where the acceleration is directly proportional to the displacement and opposite in direction. Understanding SHM allows one to predict the time-dependent behavior of the mass’s motion, including its period and frequency.

Hooke’s Law

Hooke’s Law governs the behavior of elastic materials, stating that the force exerted by an elastic element is proportional to its extension or compression from its natural length. In the context of a spring or elastic rope, it provides the basis for calculating the restoring force acting on the mass when displaced from equilibrium. This relationship is essential for establishing the linear assumptions that lead to simple harmonic motion.

Static Equilibrium

This concept involves finding the point at which the net force on the mass is zero. In problems involving a mass attached to an elastic element, the static equilibrium position is determined by equating the downward gravitational force to the upward elastic restoring force. Determining this position is the first step in analyzing any oscillatory motion as it serves as the reference point around which vibrations occur.

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