Problem 51 - University Physics with Modern Physics

Question

An \(80.0-\mathrm{kg}\) 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 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 ascillation, and c) the time for the amplitude to decrease to \(2.00 \mathrm{~m}\) (with air resistance providing the damping of the oscillations at \(7.50 \mathrm{~kg} / \mathrm{s})\)

Step-by-Step Solution

Answer

Question: A bungee jumper with a mass of 80.0 kg experiences simple harmonic motion with a period of 5.00 s and an initial amplitude of 10.0 m. The damping rate due to air resistance is 7.50 kg/s. Calculate a) the spring constant of the bungee cord, b) the maximum speed of the jumper during oscillation, and c) the time taken for the amplitude to reduce to 2.00 m. Answer: a) The spring constant is approximately 200 N/m. b) The maximum speed of the jumper during oscillation is approximately 12.6 m/s. c) The time taken for the amplitude to reduce to 2.00 m is approximately 23.5 s.

Step 1: Part a: Finding the spring constant

Using the formula for the period of oscillation: \(T = 2\pi\sqrt{\frac{m}{k}}\), the spring constant (k) can be found by rearranging the equation and solving for k. Given that the period \(T = 5.00s\) and the mass \(m = 80.0kg\), we can plug these values into our equation and solve for k: \(k = \frac{4\pi^2 m}{T^2} = \frac{4\pi^2 (80.0)}{(5.00)^2}\) The value we get for the spring constant, k, is approximately 200 N/m.

Step 2: Part b: Finding the maximum speed

To find the maximum speed during the oscillation, we first need to find the angular frequency (\(\omega = \frac{2\pi}{T}\)). Given that the period \(T = 5.00s\), we calculate the angular frequency as: \(\omega = \frac{2\pi}{5.00s} \approx 1.26 s^{-1}\) Now, using the formula for the maximum speed (\(v_{max} = A\omega\)) and given that the amplitude \(A = 10.0m\), we can calculate the maximum speed as: \(v_{max} = (10.0m)(1.26 s^{-1}) \approx 12.6 m/s\)

Step 3: Part c: Time for amplitude to decrease to 2.00m

Given that air resistance provides damping with the rate 7.50 kg/s, the amplitude's decay with time can be described by the following equation: \(A(t) = A_0 e^{-\frac{b}{2m}t}\) Here, \(A_0 = 10.0m\), \(A(t) = 2.00m\), \(b = 7.50kg/s\), and \(m = 80.0kg\). We need to solve for the time \(t\). Rearrange the equation and solve for time: \(t = \frac{-2m\ln{\frac{A(t)}{A_0}}}{b} = \frac{-2(80.0)\ln{\frac{2.00}{10.0}}}{7.50}\) The time it takes for the amplitude to decrease to 2.00m is approximately 23.5s.

Key Concepts

Spring Constant

The spring constant, commonly represented as \( k \), is a measure of the stiffness of a spring. In simple harmonic motion, it defines how much force is necessary to stretch or compress the spring by a unit length. Essentially, a stiffer spring has a higher spring constant and requires more force to deform.

For the bungee jumper problem, to find the spring constant of the bungee cord, we use the formula for the period of oscillation: \[ T = 2\pi\sqrt{\frac{m}{k}} \]Where:

• \( T \) is the period of oscillation
• \( m \) is the mass of the object attached to the spring
• \( k \) is the spring constant

To solve for the spring constant \( k \), we rearrange this formula:\[ k = \frac{4\pi^2 m}{T^2} \]Plugging in the given values \( T = 5.00\, \text{s} \) and \( m = 80.0\, \text{kg} \), we find that \( k \approx 200\, \text{N/m} \). This tells us that the cord is fairly stiff, maintaining the oscillation period when swinging the jumper back and forth.

Amplitude

Amplitude \( A \) in simple harmonic motion refers to the maximum distance from the equilibrium position during the oscillation. It is an important measurement, as it represents the maximum extent of the oscillation.

In our problem, the bungee jumper's initial amplitude is 10.0 meters. This means that, during the first oscillation, the jumper moves 10.0 meters away from the resting position, before returning through it on the way to the other extreme. Amplitude gives us an idea of how far the motion will carry the object back and forth.

As time progresses, the amplitude can decrease, especially if there are damping forces, such as air resistance, acting to reduce the energy of the system. In the given exercise, the challenge is to calculate the time it will take for the amplitude to decrease from 10.0 meters to 2.0 meters in the presence of damping. This can be described using the decay equation:\[ A(t) = A_0 e^{-\frac{b}{2m}t} \]Where:

• \( A_0 \) is the initial amplitude
• \( b \) is the damping coefficient
• \( m \) is the mass
• \( t \) is the time elapsed

This equation allows us to understand and predict how quickly the motion will naturally decrease over time.

Oscillation Period

The oscillation period, noted as \( T \), is the time it takes for one complete cycle of oscillation. This cycle includes a full pass from maximum displacement on one side to the maximum on the opposite side and back. The period gives us a scope of how fast or slow the oscillations are occurring.

In simple harmonic motion, the period is crucial because it helps determine other characteristics, such as angular frequency and maximum speed. The equation linking period and the spring constant is:\[ T = 2\pi\sqrt{\frac{m}{k}} \]For our exercise, with \( T = 5.00\, \text{s} \), the period is used to calculate the angular frequency \( \omega = \frac{2\pi}{T} \). Knowing \( T \), we can determine \( \omega \approx 1.26 \text{ s}^{-1} \), which subsequently enables us to find maximum speed with \( v_{max} = A\omega \). Each of these values plays a part in predicting the behavior of the bungee jumper as they undergo oscillations.

Damping in Oscillations

Damping in oscillations refers to forces, like friction or air resistance, that reduce the amplitude of motion over time. This process results in energy being lost from the system, often as thermal energy, causing the oscillations to gradually decrease and eventually stop.

In the bungee jumper scenario, damping is introduced as air resistance, having a damping coefficient \( b = 7.50\, \text{kg/s} \). Damping rate affects how quickly the amplitude diminishes, and it can be expressed in the equation:\[ A(t) = A_0 e^{-\frac{b}{2m}t} \]Where:

• \( A_0 \) is the initial amplitude
• \( b \) is the damping constant
• \( m \) is the mass
• \( t \) is time

By rearranging and solving for \( t \), we find how long it takes for the system to decline to a certain amplitude. Understanding damping is essential for predicting the longevity and behavior of oscillatory systems in the real world.