Question

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 \(7.50 \mathrm{~kg} / \mathrm{s}\) ).

Short answer

#tag_title#Step 2: Calculate the maximum speed of the jumper#tag_content#To find the maximum speed of the jumper during the oscillation, we can use the equation for the speed of an object in simple harmonic motion: \(v_{max} = A \omega\) Where \(v_{max}\) is the maximum speed, \(A = 12.0 m\) is the amplitude, and \(\omega\) is the angular frequency. We can find the angular frequency using the relationship between period and angular frequency: \(\omega = \frac{2 \pi}{T}\) Then, substitute the values and calculate the maximum speed. #tag_title#Step 3: Account for the air resistance#tag_content#Now that we have the maximum speed, we can take air resistance into account. We are given that it takes \(40.0 s\) for the amplitude to decrease by a factor of \(2\), which implies an exponential decay in amplitude. The equation for the amplitude as a function of time is: \(A(t) = A_0 e^{-\alpha t}\) Where \(A(t)\) is the amplitude at time \(t\), \(A_0 = 12.0 m\) is the initial amplitude, and \(\alpha\) is the decay constant. To find the decay constant, set \(A(t) = \frac{A_0}{2}\) at \(t = 40.0 s\): \(e^{-\alpha \cdot 40.0} = \frac{1}{2}\) Solve for \(\alpha\): \(\alpha = \frac{ln(2)}{40.0}\) Now that we have the decay constant, we can calculate the amplitude at any time and find the speed of the jumper accordingly. These steps allow us to analyze the oscillations of the bungee jumper, calculate their maximum speed considering the spring constant, and account for air resistance to determine how the amplitude changes over time.

Step-by-step solution

Calculate the spring constant using the period and mass of the jumper

To calculate the spring constant, we first need the angular frequency of the oscillations, which we can find using the period. The relationship between the period, mass, and spring constant is given by: \(T = 2 \pi \sqrt{\frac{m}{k}}\) Where \(T = 5.00 s\) is the period, \(m = 80.0 kg\) is the mass, and \(k\) is the spring constant. We can solve for \(k\): \(k = \frac{4 \pi^2 m}{T^2}\)

Key concepts

Oscillations

Oscillations refer to the repetitive movement of an object between two positions or states. Imagine a child on a swing, moving back and forth; this movement is characteristic of an oscillation. In physics, oscillations can describe many types of periodic motions, from the macroscopic, like the pendulum of a clock, to the microscopic, such as the vibrations of atoms.

One of the familiar properties of oscillations is amplitude—the maximum displacement from the rest position. In our exercise, the bungee jumper's initial amplitude is 10.0 meters. Period, another key property, is the time taken for one complete cycle of the motion, which for the jumper is 5.00 seconds. Understanding these concepts is crucial for grasping the physics of oscillatory systems and solving problems related to them.

Harmonic Motion

Harmonic motion is a type of oscillation where the restoring force is directly proportional to the displacement but in the opposite direction. This kind of motion is predictable and is described by sine or cosine functions mathematically. For instance, when a spring is stretched or compressed, it exerts a force to return to its original length. This is a classic example of harmonic motion called simple harmonic motion (SHM).

In the context of the bungee jumper, if we consider the bungee cord as an ideal spring, it exhibits SHM when the jumper oscillates up and down. SHM can be characterized by its periodicity and uniformity, meaning objects in SHM will take the same amount of time to complete each oscillation, provided the system doesn't change.

Damping

Damping refers to the influence of an external force, such as friction or air resistance, which gradually reduces the energy of an oscillating system, causing the amplitude of the oscillations to decrease over time. Most real-world oscillating systems experience some form of damping.

In our bungee jumping example, the exercise asks to consider the damping effect of air resistance. The provided damping coefficient is 7.50 kg/s, which signifies the rate at which the amplitude of oscillation diminishes. While the initial problem assumes no damping for simplicity, the subsequent question acknowledges the realistic situation where damping forces can't be ignored.

Angular Frequency

Angular frequency is fundamental to the study of oscillations and represents the rate of change of the phase angle in harmonic motion. The angular frequency, usually denoted by the Greek letter omega (\r\(\omega\)), is measured in radians per second. It's related to the frequency (\r\(f\)), which is the number of oscillations per second, by the equation \r\(\omega = 2\pi f\).

The angular frequency for a mass-spring system (like our bungee cord) is also given by \r\(\omega = \sqrt{\frac{k}{m}}\), where \r\(k\) is the spring constant, and \r\(m\) is the mass. In the exercise, we use the period (the inverse of frequency) to solve for the spring constant through a formula involving angular frequency, illustrating the intertwined relationship between these concepts.