An 80.0 -kg bungee jumper is enjoying an afternoon of jumps....

An 80.0 -kg bungee jumper is enjoying an afternoon of jumps. The jumper's first oscillation has an amplitude of $10.0 \mathrm{~m}$ and a period of $5.00 \mathrm{~s}$. Treating the bungee cord as a spring with no damping, calculate each of the following: a) the spring constant of the bungee cord, b) the bungee jumper's maximum speed during the oscillation, and c) the time for the amplitude to decrease to $2.00 \mathrm{~m}$ (with air resistance providing the damping of the oscillations at $7.50 \mathrm{~kg} / \mathrm{s}$ ).

An 80.0 -kg bungee jumper is enjoying an afternoon of jumps. The jumper's first oscillation has an amplitude of $10.0 \mathrm{~m}$ and a period of $5.00 \mathrm{~s}$. Treating the bungee cord as a spring with no damping, calculate each of the following:
a) the spring constant of the bungee cord,
b) the bungee jumper's maximum speed during the oscillation, and
c) the time for the amplitude to decrease to $2.00 \mathrm{~m}$ (with air resistance providing the damping of the oscillations at $7.50 \mathrm{~kg} / \mathrm{s}$ ).

Key Concepts

Damped Harmonic Motion

Real-world oscillators often experience damping, where non-conservative forces like air resistance cause the amplitude of oscillations to decay over time. In a damped harmonic oscillator, a damping coefficient (or damping constant) is present in the system's equations of motion, altering the behavior from the ideal, lossless scenario.

Oscillatory Motion Velocity

The maximum speed in simple harmonic motion occurs as the system passes through the equilibrium position. This speed can be determined using the relationship v_max = A?, where A is the amplitude and ?, the angular frequency, is calculated from the mass and the spring constant. This highlights how the system’s energy distribution influences the peak velocity during oscillations.

Exponential Decay of Oscillation Amplitude

When damping is present, the amplitude of an oscillatory system decays exponentially with time. This decay is typically modeled by the relation A(t) = A? exp(?(c/(2m)) t), where A? is the initial amplitude, c is the damping coefficient, and m is the mass. This concept allows one to predict how long it will take for the amplitude to decrease to a specified value.

Simple Harmonic Motion

This concept describes the oscillatory motion where the restoring force is directly proportional to the displacement from an equilibrium position and is directed towards that equilibrium position. It is central to analyzing systems like springs and pendulums, where the motion repeats in a sinusoidal manner, characterized by parameters such as amplitude, period, and angular frequency.

Determination of the Spring Constant

In systems modeled as a spring, such as a bungee cord in a simplified case, the spring constant (k) quantifies the stiffness of the spring. It is directly related to the period of oscillation and the mass of the object via the formula T = 2??(m/k). This relationship allows for calculating k when the period of oscillation and the mass are known.

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