Problem 65 - College Physics

Question

\(\bullet$$\bullet\) You've attached a bungee cord to a wagon and are using it to pull your little sister while you take her for a jaunt. The bungee's unstretched length is \(1.3 \mathrm{m},\) and you happen to know that your little sister weighs 220 \(\mathrm{N}\) and the wagon weighs 75 \(\mathrm{N}\) . Crossing a street, you accelerate from rest to your normal walk- ing speed of 1.5 \(\mathrm{m} / \mathrm{s}\) in 2.0 \(\mathrm{s}\) , and you notice that while you're accelerating, the bungee's length increases to about 2.0 \(\mathrm{m}\) . What's the force constant of the bungee cord, assuming it obeys Hooke's law?

Step-by-Step Solution

Answer

Approximately 32.25 N/m.

Step 1: Identify known quantities

We need to solve for the force constant (spring constant) of the bungee cord. We are given the following:- Unstretched length of the bungee, \(L_0 = 1.3\, \text{m}\)- Stretched length of the bungee, \(L = 2.0\, \text{m}\)- Weight of the sister, \(f_{sister} = 220\, \text{N}\)- Weight of the wagon, \(f_{wagon} = 75\, \text{N}\)- Final velocity, \(v_f = 1.5\, \text{m/s}\)- Time to reach this velocity, \(t = 2.0\, \text{s}\)

Step 2: Calculate total mass

First, calculate the total weight being pulled by the bungee, which is the sum of the sister's and the wagon's weights:\[f_{total} = f_{sister} + f_{wagon} = 220\, \text{N} + 75\, \text{N} = 295\, \text{N}\]Convert the total weight to mass (remember, weight \(=\) mass \(\times\) gravity):\[m_{total} = \frac{f_{total}}{g} = \frac{295\, \text{N}}{9.8\, \text{m/s}^2} \approx 30.1\, \text{kg}\]

Step 3: Calculate acceleration

We know the initial velocity \(v_i = 0\, \text{m/s}\), the final velocity \(v_f = 1.5\, \text{m/s}\), and the time \(t = 2.0\, \text{s}\). Use the formula for acceleration:\[a = \frac{v_f - v_i}{t} = \frac{1.5\, \text{m/s} - 0}{2.0\, \text{s}} = 0.75\, \text{m/s}^2\]

Step 4: Calculate force due to acceleration

Apply Newton's second law to find the force required to accelerate the mass at this rate:\[F = m_{total} \times a = 30.1\, \text{kg} \times 0.75\, \text{m/s}^2 = 22.575\, \text{N}\]

Step 5: Calculate force constant of the bungee cord

The extension of the bungee cord is the difference between its stretched length and its unstretched length:\[\Delta L = L - L_0 = 2.0\, \text{m} - 1.3\, \text{m} = 0.7\, \text{m}\]Using Hooke's Law, \(F = k \times \Delta L\), where \(k\) is the force constant we're solving for:\[k = \frac{F}{\Delta L} = \frac{22.575\, \text{N}}{0.7\, \text{m}} \approx 32.25\, \text{N/m}\]

Step 6: Final Result

The force constant of the bungee cord, given the described stretch and acceleration conditions, is approximately \(32.25\, \text{N/m}\).

Key Concepts

Newton's second law

This fundamental principle of physics states that the force applied to an object is equal to the mass of that object multiplied by its acceleration. Newton's second law is mathematically expressed as:

• \( F = m \cdot a \),

where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration.
In our exercise, the little sister and her wagon represent the object being accelerated. By using this law, we calculate the necessary force needed to bring them to a speed of 1.5 m/s in a 2-second timespan. Understanding how force is tied to mass and acceleration is pivotal when analyzing dynamic systems like a bungee cord in motion.

Hooke's law

Hooke's law describes the behavior of springs and other elastic materials. It relates the force exerted by a spring to its extension or compression from the original length. The formula according to Hooke’s law is:

• \( F = k \cdot \Delta L \),

where \( F \) is the force exerted by the spring, \( k \) is the spring constant or force constant, and \( \Delta L \) is the change in the spring's length.
In the context of the bungee cord, the unstressed length was 1.3 m and it stretched to 2.0 m when being pulled. By using Hooke's law, we find the spring constant \( k \) by dividing the force exerted by the extension of the bungee.

Acceleration calculation

Acceleration is the rate of change of velocity of an object. When you increase your speed from rest to a walking pace, you experience acceleration. This exercise asks us to determine the acceleration by using the formula:

• \( a = \frac{v_f - v_i}{t} \),

where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( t \) is the time taken to change from \( v_i \) to \( v_f \).
In this case, the wagon and sister started from rest, so \( v_i = 0 \) m/s, reached a speed of 1.5 m/s in 2 seconds. This calculation revealed the acceleration as 0.75 \( \text{m/s}^2 \). Understanding this concept is crucial when examining how quickly something changes motion.

Force constant

The force constant, or spring constant, \( k \), measures the stiffness of a spring or elastic material. A higher force constant means the material is stiffer and harder to stretch or compress.
In the bungee cord scenario, we determined the force constant by imagining the exertion needed to stretch the cord. By dividing the force (which we found using Newton’s second law) by the cord's extension (given by the difference in original and stretched lengths), we deduce the force constant of the bungee as approximately 32.25 \( \text{N/m} \). This tells us how much force is needed per meter of extension.

Bungee cord dynamics

The dynamics of a bungee cord are all about how it stretches and recoils back in response to forces, governed by laws like Newton’s and Hooke’s. Bungee cords are popular in activities for their elasticity, where they must be strong enough to support weight yet elastic enough to absorb shocks and provide bounce.

• The unstretched length gives us a baseline to measure how far the cord extends under force.
• The force constant gives insight into the cord's stiffness.
• An understanding of acceleration further explains how quickly a weight can shift these physical states.

By combining all these calculations, we can predict the cord's behavior under different weights and movements, which is essential for safe and thrilling bungee activities.