Problem 51 - University Physics with Modern Physics

Question

Bungee Jump. A bungee cord is 30.0 \(\mathrm{m}\) long and, when stretched a distance \(x,\) it exerts a restoring force of magnitude \(k x\) Your father-in- law (mass 95.0 \(\mathrm{kg}\) ) stands on a platform 45.0 \(\mathrm{m}\) above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 \(\mathrm{m}\) before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 380.0 \(\mathrm{N}\) . When you do this, what distance will the bungee cord that you should select have stretched?

Step-by-Step Solution

Answer

The bungee cord should stretch about 4.49 m when a 380 N force is applied.

Step 1: Understanding the Problem

A bungee cord applies a restoring force given by Hooke's Law, which states that the force exerted by a spring is proportional to its extension or compression, expressed as \( F = kx \). Here, \( k \) is the spring constant, \( x \) is the extension, and \( F \) the force applied. We need to select a cord such that a person drops without hitting the ground, considering a fall from 45 m to a maximum of 41 m.

Step 2: Calculate Gravitational Force

The gravitational force acting on the person is their weight, calculated as \( F_g = mg \), where \( m = 95.0 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). Therefore, \( F_g = 95.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 931.0 \, \text{N} \).

Step 3: Elongation Beyond Natural Length

The total fall is 41 m, but only the excess of 11 m beyond the natural cord length of 30 m stretches the bungee cord. So, \( x = 11 \, \text{m} \).

Step 4: Determining Spring Constant \(k\)

Using Hooke's Law, \( F = kx \). Plugging in \( F_g = 931 \, \text{N} \) and \( x = 11 \, \text{m} \), we get \( k = \frac{931}{11} \approx 84.64 \, \text{N/m} \).

Step 5: Testing the Bungee Cord with a Force

A test force of 380.0 N stretches the cord. Using the calculated \( k \) value, the extension \( x \) is found by rearranging Hooke's Law: \( x = \frac{380}{84.64} \approx 4.49 \, \text{m} \).

Key Concepts

Hooke's Law

Hooke's Law is a fundamental concept used to understand how springs and elastic materials behave when they are stretched or compressed. It states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed. This relationship is mathematically described by the equation: \[ F = kx \] where:

• \( F \) is the force exerted by the spring (in Newtons),
• \( k \) is the spring constant (in Newtons per meter), representing the stiffness of the spring, and
• \( x \) is the displacement of the spring from its natural length (in meters).

In the context of a bungee jump, Hooke’s Law helps in determining how much a bungee cord will stretch when a force, such as the weight of the jumper, is applied to it. Understanding this concept is crucial in ensuring safety and selecting the right bungee cord that will absorb the fall without letting the jumper hit the ground.

Spring Constant

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. In practical terms, it tells us how much force is needed to stretch or compress the spring by a unit length. In our bungee jump problem, determining the correct spring constant is critical to ensure that the bungee cord stretches just enough to stop the jumper safely before hitting the ground.
The formula for the spring constant \( k \) is derived from Hooke’s Law: \[ k = \frac{F}{x} \]where:

• \( F \) is the force applied on the spring,
• \( x \) is the stretch or compression of the spring.

For example, if a person weighing 931 N stretches the cord by 11 meters while falling, the spring constant would be \( k = \frac{931}{11} \approx 84.64 \, \text{N/m} \). This calculation helps in selecting a bungee cord that will extend correctly when the person jumps.

Force Calculations

Force calculations play a pivotal role in designing a safe bungee jump. It involves calculating the various forces acting on the jumper during the fall. The two primary forces to focus on are the gravitational force and the restoring force exerted by the bungee cord.
To calculate the forces involved:

• The gravitational force is determined by the jumper's mass and the acceleration due to gravity: \[ F_g = mg \]where \( m \) is the mass and \( g \) is the gravitational acceleration (approximately \( 9.8 \, \text{m/s}^2 \)).
• The restoring force exerted by the bungee cord is calculated using Hooke’s Law: \[ F = kx \]

By ensuring both forces are correctly calculated, one can ensure that the chosen bungee cord will effectively slow down and stop the jumper at the right point.

Gravitational Force

Gravitational force is the force of attraction between objects with mass. For a bungee jump, it is the force that pulls the jumper towards the Earth. Calculating gravitational force is crucial when determining how much a bungee cord will stretch.
Using the formula:\[ F_g = mg \]we can find the gravitational force acting on the jumper. Here:

• \( F_g \) is the gravitational force,
• \( m \) is the jumper’s mass in kilograms,
• \( g \) is the acceleration due to gravity, which is approximately \( 9.8 \, \text{m/s}^2 \).

For instance, if the jumper weighs 95 kg, the gravitational force would be \( 95 \times 9.8 = 931 \, \text{N} \). This force is what initially accelerates the jumper as they fall before the bungee cord begins to stretch and halt the descent.

Physics Problem Solving

Solving physics problems like the bungee jump scenario requires a structured approach. It involves identifying given information, applying relevant concepts, and performing calculations to find solutions. Here are the key steps:

• Understand the Problem: Identify what is being asked and the applicable physics principles, such as Hooke's Law.
• Identify Known and Unknown Variables: Determine what information is provided, like the mass and fall distance, and what needs to be calculated, like the spring constant \( k \).
• Apply Physics Equations: Use the relevant equations, such as \( F = kx \) for restoring force and \( F_g = mg \) for gravitational force.
• Perform Calculations: Solve the equations using the given data to find unknowns like the stretch length or the spring constant.
• Review the Solution: Check if the results make sense and ensure that safety constraints, like stopping before hitting the ground, are met.

A strategic approach not only leads to the correct solution but also deepens the understanding of the physics involved.