Problem 15 - Physics for Scientists and Engineers with Modern Physics

Question

(1I) A 55 -kg bungee jumper leaps from a bridge. She is tied to a bungee cord that is 12 \(\mathrm{m}\) long when unstretched, and falls a total of 31 \(\mathrm{m}\) . (a) Calculate the spring constant \(k\) of the bungee cord assuming Hooke's law applies. (b) Calcu- late the maximum acceleration she experiences.

Step-by-Step Solution

Answer

The spring constant \( k \) is approximately 57.118 N/m, and the maximum acceleration is about 19.721 m/s².

Step 1: Understanding the Problem

We are given a bungee jumper of mass 55 kg, who jumps from a bridge with a bungee cord that is 12 m long when unstretched. The jumper falls a total of 31 m. We need to find the spring constant \( k \) of the bungee cord, and the maximum acceleration the jumper experiences.

Step 2: Define Key Variables

Let the mass \( m = 55 \text{ kg} \), the length of the unstretched bungee \( L_0 = 12 \text{ m} \), and the total fall distance \( d = 31 \text{ m} \). The stretch of the bungee cord is \( x = d - L_0 = 31 - 12 = 19 \text{ m} \).

Step 3: Apply Energy Conservation

Assuming conservation of mechanical energy, the potential energy lost equals the elastic potential energy gained at maximum stretch. Thus, \( mgh = \frac{1}{2}kx^2 \). Solve for the spring constant \( k \): \[ k = \frac{2mgh}{x^2} \]Substitute \( h = 19 \text{ m} \) since the cord stretches 19 m, and \( g = 9.81 \text{ m/s}^2 \).

Step 4: Calculate the Spring Constant \( k \)

Substitute known values into the equation for \( k \): \[ k = \frac{2 \times 55 \times 9.81 \times 19}{19^2} \approx 57.118 \text{ N/m} \].

Step 5: Find Maximum Acceleration

According to Hooke's law, the force exerted by the bungee cord is \( F = kx \). The maximum force occurs when the bungee is at maximum stretch, \( F = 57.118 \times 19 \). Use Newton's second law \( F = ma \) to find the maximum acceleration \( a \):\[ a = \frac{F}{m} = \frac{57.118 \times 19}{55} \approx 19.721 \text{ m/s}^2 \].

Key Concepts

Spring Constant Calculation

Spring constant, often symbolized as \( k \), is a fundamental parameter in bungee jumping physics. It represents the stiffness of the bungee cord. A higher spring constant indicates a stiffer cord.
To calculate it, use the principles of energy conservation. When the bungee cord stretches, gravitational potential energy is converted into elastic potential energy. This can be expressed as:

• Gravitational potential energy: \( mgh \)
• Elastic potential energy: \( \frac{1}{2}kx^2 \)

Therefore, using energy conservation: \( mgh = \frac{1}{2}kx^2 \). Solving for \( k \), we have \[ k = \frac{2mgh}{x^2} \]. This formula allows you to calculate the spring constant if you know the mass, gravitational acceleration, and the stretch distance. In our example, the calculated spring constant is approximately 57.118 N/m.

Hooke's Law

Hooke's Law is crucial for understanding the behavior of bungee cords. It states that the force exerted by a spring is proportional to its extension or compression, within the elastic limit of the material. Mathematically, it is expressed as: \[ F = kx \]

• \( F \) is the force exerted by the spring, in newtons (N).
• \( k \) is the spring constant, in N/m.
• \( x \) is the displacement from the equilibrium position, in meters (m).

Hooke's Law helps predict how a bungee cord will react when stretched during a jump. It works well until the material reaches its elastic limit, beyond which it may not return to its original shape. In our case, we see that the force exerted by the bungee cord increases as it stretches further.

Energy Conservation in Physics

Energy conservation is a pivotal concept in physics, particularly in activities like bungee jumping. It states that the total energy in a closed system remains constant, though it can change forms.
For the bungee jumper:

• Initially, the jumper has gravitational potential energy: \( mgh \).
• As they fall, this energy converts to kinetic energy and, eventually, elastic potential energy at full stretch: \( \frac{1}{2}kx^2 \).

By setting these equations equal, we derive the spring constant. This concept shows how the jumper's potential energy is never lost but converted into other types of energy, allowing them to return upwards.

Newton's Second Law

Newton's Second Law, which states \( F = ma \), allows us to calculate the maximum acceleration experienced by the bungee jumper. It describes how the velocity of an object changes when it is subjected to an external force. In simpler terms, the force applied to an object and the acceleration it undergoes are directly proportional, considering that its mass remains constant.

• \( F \) is the net force acting on the object.
• \( m \) is the object's mass.
• \( a \) is the acceleration.

For the jumper, the maximum force from the bungee's stretch can be found using Hooke's Law, and then plugged into Newton's Second Law to find maximum acceleration: \[ a = \frac{F}{m} \]. In our scenario, this calculation tells us that the jumper experiences a maximum acceleration of around 19.721 m/s² at the lowest point of her descent.