Problem 47 - University Physics with Modern Physics

Question

A bungee cord is 30.0 m long and, when stretched a distance \(x\), it exerts a restoring force of magnitude \(kx\). Your father-in-law (mass 95.0 kg) stands on a platform 45.0 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 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 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 around 11 meters when pulled with 380 Newtons.

Step 1: Understand the Problem

The total fall distance should not exceed 41.0 m. The bungee cord initially has a length of 30.0 m. Thus, the maximum stretching of the bungee cord alone is 11.0 m (41.0 m - 30.0 m). The restoring force of the cord is proportional to the stretch length, given by Hooke's Law: \( F = kx \). We need to find the stretch length \( x \) for a restoring force of 380.0 N.

Step 2: Find the Spring Constant

Given \( F = kx \), we rearrange this to \( k = \frac{F}{x} \). We need this equation to eventually ensure that maximum force doesn't exceed the gravitational force exerted by your father-in-law's mass at maximum stretch.

Step 3: Use Hooke's Law and the Conditions

For equilibrium, the restoring force should balance with the gravitational force when the cord is fully stretched. The gravitational force when the cord is fully stretched is \( F = mg = 95 \times 9.81 \). Let this \( F \) be equal to the restoring force of 380 N.

Step 4: Solve for the Stretched Length

Using Hooke's Law, we have \( F = kx \). Plug in the numbers: \( 380 = k \cdot 11 \). Solving for \( x \), we have \( x = \frac{380}{k} \). Substitute back, knowing \( F = mg \), and solve using the condition where \( F = 931.95 \) N at maximum stretch.

Step 5: Calculate the Needed Stretch

Since \( x = \frac{380}{k} \) and \( k \approx 34.54 \) N/m, solving for \( x \) in the test scenario gives: \( x = \frac{380}{34.54} \). Calculate \( x \) for this scenario.

Key Concepts

Restoring Force

Restoring force is a fundamental concept when dealing with elastic materials, like bungee cords, and is, in essence, the force that acts to bring an object back to its original shape or position. This is perfectly explained by Hooke's Law, which states that the force required to extend or compress a spring by some distance is proportional to that distance. Hence, the formula is expressed as:

• \( F = kx \), where \( F \) is the restoring force, \( k \) is the spring constant, and \( x \) is the displacement from its original length.

The restoring force acts in the opposite direction to the displacement of the bungee cord, ensuring that once stretched, it pulls back to its unstressed state. This can be a challenging concept at first! But remembering that the restoring force always strives to bring the object back to equilibrium will help you solve problems related to bungee jumping and other spring-related activities.

Bungee Cord

A bungee cord is an elastic rope that is often used in activities like bungee jumping. Its primary feature is elasticity, allowing it to stretch under force and then return to its original shape. In the given problem, the bungee cord stretches a certain distance when your father-in-law jumps. Initially, the bungee cord is 30.0 meters long. When the maximum fall distance is set at 41.0 meters, the elastic portion (when the cord is stretched) is 11.0 meters long. Here’s how this plays out:

• Unstretched Length: 30.0 m
• Maximum Length: 41.0 m
• Stretch Length: 11.0 m (41.0 m - 30.0 m)

The elasticity allows it to safely bring someone back to rest after a jump. Just like in the problem, a well-stretched bungee cord can prevent your father-in-law from hitting the ground by stopping the fall after 41 meters.

Spring Constant

The spring constant, often denoted by \( k \), is a measure of how stiff or rigid a spring or elastic material like a bungee cord is. This value determines how much force is needed to stretch or compress the object by a unit length. It is pivotal in understanding how the bungee cord behaves under stress.Using Hooke's Law, \( F = kx \), the spring constant can be derived as:

• \( k = \frac{F}{x} \)

In the exercise given, we calculated the spring constant using a force of 380 N, required to stretch the test bungee, and an unknown \( x \). This constant helps ensure the cord doesn't overstretch dangerously with a weight as significant as your father-in-law's.For calculations: the gravitational force when jumping (using mass \( 95 \) kg and \( g \) as \( 9.81 \) m/s²) helps ensure the spring constant withstands the actual force during the jump while ensuring safety.

Physics Problem-Solving

Solving physics problems requires a systematic approach to ensure all aspects and conditions are met accurately. In this bungee jump problem, multiple steps ensure the right answers. Key steps involve:

• Understanding the Problem: Identify what is known (initial cord length and max safe fall distance) and unknown (stretch in test scenario).
• Using Relevant Equations: Apply Hooke’s Law to work with the forces involved, particularly stressing how the restoring force balances with gravitational force.
• Incorporating Constraints: Gravitational force, represented as \( F = mg \), where \( m \) is mass and \( g \) is gravity, must not be exceeded.
• Solving with Context: Test the bungee corde with a known weight, ensuring theoretical and real conditions align for safety.

Mastering these techniques requires practice, patience, and a clear understanding of each component involved, ultimately leading to an accurate and safe solution.