Problem 57 - College Physics

Question

An apple weighs \(1.00 \mathrm{~N}\). When you hang it from the end of a long spring of force constant \(1.50 \mathrm{~N} / \mathrm{m}\) and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back- and-forth swings do not cause any appreciable change in the length of the spring.) What is the unstretched length of the spring (with the apple removed)?

Step-by-Step Solution

Answer

The unstretched length of the spring is approximately 1.98 m.

Step 1: Analyze the problem

We are given a problem involving an apple hanging on a spring, making it a case of simple harmonic motion (SHM). We need to determine the unstretched length of the spring. The frequency of the apple bouncing vertically in SHM is double the frequency when it swings like a pendulum.

Step 2: Calculate the bounce frequency

The angular frequency of the apple bouncing up and down is given by the formula for a mass-spring system:\[ \omega = \sqrt{\frac{k}{m}} \]where \( k = 1.50 \, \text{N/m} \) is the spring constant and \( m \) is the mass of the apple. We know the weight \( W = 1.00 \, \text{N} \), thus \( m = \frac{W}{g} = \frac{1.00 \, \text{N}}{9.81 \, \text{m/s}^2} \approx 0.102 \text{kg} \). The angular frequency is therefore:\[ \omega = \sqrt{\frac{1.50}{0.102}} \approx 3.85 \, \text{rad/s} \]

Step 3: Find the bounce frequency

The frequency in hertz \( f_b \) is given by:\[ f_b = \frac{\omega}{2\pi} = \frac{3.85}{2\pi} \approx 0.613 \, \text{Hz} \]

Step 4: Determine the pendulum frequency

According to the problem, the pendulum frequency \( f_p \) is half of the bounce frequency:\[ f_p = \frac{f_b}{2} = \frac{0.613}{2} \approx 0.307 \, \text{Hz} \]

Step 5: Use the pendulum frequency to find the effective length

The frequency of a simple pendulum is given by:\[ f_p = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]Solving for \( L \), the effective length of the pendulum (which initially is impacted by the spring's stretch), we have:\[ L = \frac{g}{(2\pi f_p)^2} \]Substituting \( g = 9.81 \, \text{m/s}^2 \) and \( f_p = 0.307 \, \text{Hz} \), we find:\[ L = \frac{9.81}{(2\pi \times 0.307)^2} \approx 2.65 \, \text{m} \]

Step 6: Find the spring's stretch with the apple

The spring stretch \( \Delta x \) with the apple hanging is given by Hooke's law:\[ F = k \Delta x \Rightarrow \Delta x = \frac{F}{k} = \frac{1.00}{1.50} \approx 0.667 \, \text{m} \]

Step 7: Calculate the unstretched length

The effective length of the pendulum \( L \) includes both the unstretched spring and the stretch due to the apple. Hence, the unstretched length \( L_0 \) is:\[ L_0 = L - \Delta x = 2.65 - 0.667 \approx 1.98 \, \text{m} \]

Key Concepts

Mass-Spring System

In physics, the mass-spring system is a fundamental model used to understand simple harmonic motion (SHM). Imagine hanging an object, like an apple, on a spring. When you pull the apple down and release it, the apple bounces up and down in a rhythmic pattern. This pattern is termed as simple harmonic motion.

The motion arises due to the spring's tendency to return to its original shape when stretched or compressed. The system involves a mass attached to a spring with a certain spring constant, and its behavior can be understood through equations of motion.

• The system oscillates with a characteristic frequency, determined by the mass of the object and the stiffness (spring constant) of the spring.
• The mass provides inertia, while the spring provides a restoring force.
• This results in a motion that repeats itself, making it a prime example of periodic motion.

Bouncing starts with potential energy, converting to kinetic energy when the spring moves, and back again as it reaches its maximum extension.

This energy interplay is key to understanding the dynamics of SHM.

Pendulum Frequency

When looking beyond the vertical bounce, consider the apple swinging side to side like a pendulum. This motion also displays simple harmonic characteristics if the angle is small.

Pendulums have their own unique frequency, which can be understood by considering the length of the pendulum and the gravitational pull acting on it.

• The pendulum swings back and forth because of Earth's gravity pulling it downward.
• The frequency of a simple pendulum can be affected by the length of the pendulum and the acceleration due to gravity.
• It’s important to note that the amplitude, or how far the pendulum swings, doesn’t affect the frequency as long as the swings are small.

The frequency of a pendulum is calculated using the formula:

\[ f = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]

Where \( f \) is the frequency, \( g \) is the acceleration due to gravity, and \( L \) is the length of the pendulum. For our apple, this calculation helps relate the vertical bounce to the side swing frequency.

Spring Constant

The spring constant, denoted by \( k \), is a measure of a spring's stiffness. A large spring constant means that a spring is very stiff, and a small constant indicates a looser or more flexible spring. This constant plays a vital role in determining the behavior of the mass-spring system.

In the context of our apple, we use the spring constant to determine how much the spring stretches under the apple's weight. This stretch is central to understanding how the system behaves when displaced from equilibrium.

• It dictates how much force is necessary to stretch or compress the spring by a particular distance.
• This relationship is captured by Hooke’s law: \( F = k \Delta x \), where \( F \) is the force applied, and \( \Delta x \) is the displacement from equilibrium.
• Spring constants can vary depending on the spring's material and design, making precise calculations crucial for accurate predictions.

Understanding spring constants is essential for solving problems involving elastic potential energy and oscillations.

Hooke's Law

Hooke's law forms the foundation for understanding how elastic materials, like springs, behave under load. This principle states that the force required to extend or compress a spring by some distance is proportional to that distance. The formula is elegantly simple:

\[ F = k \Delta x \]

Here, \( F \) represents the force applied to the spring, \( k \) is the spring constant, and \( \Delta x \) is the change in the spring’s length from its natural length.

• It helps us predict how far a spring will stretch or compress under a specific force.
• This law is only applicable within the elastic limit of the spring, beyond which permanent deformation may occur.
• In our context, it allows us to calculate how much our spring stretched when the apple was attached and helps us determine the unstretched length.

Hooke's law is vital in engineering and physics. It also demonstrates how springs absorb and store energy, making them effective in various mechanical systems.