Conservation of Energy

The energy of a system is conserved.

Internal energy

A system with an internal structure can have internal energy, and changes in a system’s internal structure can result in changes in internal energy

Here are some key things to know about the internal energy of a simple harmonic oscillator:

• The internal energy of an object is the energy associated with the random, chaotic motion of its constituent particles. It is a measure of the thermal energy of the object and is often symbolized by the letter U.
• In a simple harmonic oscillator, the internal energy is stored in the form of elastic potential energy when the oscillator is displaced from its equilibrium position. As the oscillator oscillates back and forth, the internal energy of the system is continually converted into kinetic energy and back into potential energy.
• The internal energy of a simple harmonic oscillator is periodic, meaning that it follows a repeating pattern over time. The internal energy of the oscillator is at a maximum when the oscillator is at its maximum displacement from its equilibrium position and is at a minimum when the oscillator is at its equilibrium position.
• The internal energy of a simple harmonic oscillator can be calculated using the equation: U = 1/2*kx^2, where U is the internal energy, k is the spring constant, and x is the displacement of the oscillator from its equilibrium position.
• The internal energy of a simple harmonic oscillator is a useful quantity to consider when analyzing the behavior of the oscillator, as it can help to understand how the energy of the system is being converted between kinetic and potential energy over time.

Potential Energy

A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces.

Here are some key things to know about the potential energy of a simple harmonic oscillator:

• The potential energy of an object is the energy that an object possesses due to its position or configuration within a force field. It is a measure of the potential for the object to do work and is often symbolized by the letter U.
• In a simple harmonic oscillator, the potential energy of the system is stored in the form of elastic potential energy when the oscillator is displaced from its equilibrium position. This energy is due to the deformation of the spring or other force-generating element in the system as it tries to return to its equilibrium position.
• The potential energy of a simple harmonic oscillator is periodic, meaning that it follows a repeating pattern over time. The potential energy of the oscillator is at a maximum when the oscillator is at its maximum displacement from its equilibrium position, and is at a minimum when the oscillator is at its equilibrium position.
• The potential energy of a simple harmonic oscillator can be calculated using the equation: U = 1/2*kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of the oscillator from its equilibrium position.
• The potential energy of a simple harmonic oscillator is a useful quantity to consider when analyzing the behavior of the oscillator, as it can help to understand how the energy of the system is being stored and converted between kinetic and potential energy over time.

Kinetic Energy 

The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the configuration of objects that make up the system.

Here are some key things to know about the kinetic energy of a simple harmonic oscillator:

• The kinetic energy of an object is the energy associated with the motion of the object. It is a measure of the ability of the object to do work due to its motion and is often symbolized by the letter K.
• In a simple harmonic oscillator, the kinetic energy of the system is stored in the form of kinetic energy when the oscillator is moving. This energy is due to the motion of the oscillator as it oscillates back and forth.
• The kinetic energy of a simple harmonic oscillator is periodic, meaning that it follows a repeating pattern over time. The kinetic energy of the oscillator is at a maximum when the oscillator is at its maximum velocity, and is at a minimum when the oscillator is at its equilibrium position or at a point of maximum displacement from the equilibrium position.
• The kinetic energy of a simple harmonic oscillator can be calculated using the equation: K = 1/2*mv^2, where K is the kinetic energy, m is the mass of the oscillator, and v is the velocity of the oscillator.
• The kinetic energy of a simple harmonic oscillator is a useful quantity to consider when analyzing the behavior of the oscillator, as it can help to understand how the energy of the system is being converted between kinetic and potential energy over time.

Energy in Simple Harmonic Oscillators

This topic is pretty much just an application of the energy types and conversions we covered in Unit 4: Energy. The main idea is that through SHM, the energy is converted from potential to kinetic and back again throughout the motion. The maximum potential energy occurs when the spring is stretched (or compressed) the most, and the maximum kinetic energy occurs at the equilibrium point. 

Image Credit

Here’s an example using a mass on a spring, resting on a frictionless surface. In pictures A, C, and E, the energy is fully stored as potential energy in the spring. In pictures B and D, the mass is at the equilibrium position (x=0) and all the energy is now kinetic energy.

If we were to make a graph of energy vs time, it would look like this:

Image Credit

A couple of things to notice in this graph above:

1. The total energy is constant. This makes sense since there are no external forces to do work on the spring-mass system
2. The potential energy and kinetic energy graphs are curves. Because of the squared term in the potential energy equation, we expect this. If the term is to the 1st power, then the graph would be linear.
3. The potential energy is greatest when the position graph is at its maximum. The Kinetic Energy is greatest when the velocity graph is at its maximum.

Example Problem 1:

A mass of 1 kg is attached to a spring with a spring constant of 50 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.2 meters from its equilibrium position and released from rest. What is the total energy of the oscillator at the maximum displacement from the equilibrium position?

Solution:

The total energy of a simple harmonic oscillator is the sum of its potential energy and kinetic energy.

The potential energy of a simple harmonic oscillator is given by the equation: U = 1/2*kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of the oscillator from its equilibrium position.

The kinetic energy of a simple harmonic oscillator is given by the equation: K = 1/2*mv^2, where K is the kinetic energy, m is the mass of the oscillator, and v is the velocity of the oscillator.

In this problem, the mass of the oscillator is 1 kg, the spring constant is 50 N/m, and the displacement from the equilibrium position is 0.2 meters.

At the maximum displacement from the equilibrium position, the velocity of the oscillator is zero and the potential energy is at a maximum.

Therefore, the total energy of the oscillator at the maximum displacement from the equilibrium position is: U + K = (1/2)(50 N/m)(0.2 m)^2 + 0 = 1 J

This means that the total energy of the oscillator at the maximum displacement from the equilibrium position is 1 J.

Example Problem 2:

A mass of 2 kg is attached to a spring with a spring constant of 100 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.5 meters from its equilibrium position and released from rest. What is the total energy of the oscillator at the equilibrium position?

Solution:

The total energy of a simple harmonic oscillator is the sum of its potential energy and kinetic energy.

The potential energy of a simple harmonic oscillator is given by the equation: U = 1/2*kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of the oscillator from its equilibrium position.

The kinetic energy of a simple harmonic oscillator is given by the equation: K = 1/2*mv^2, where K is the kinetic energy, m is the mass of the oscillator, and v is the velocity of the oscillator.

In this problem, the mass of the oscillator is 2 kg, the spring constant is 100 N/m, and the displacement from the equilibrium position is 0.5 meters.

At the equilibrium position, the velocity of the oscillator is zero and the potential energy is at a minimum.

Therefore, the total energy of the oscillator at the equilibrium position is: U + K = 1/2(100 N/m)(0.5 m)^2 + 0 = 12.5 J

This means that the total energy of the oscillator at the equilibrium position is 12.5 J.

🎥Watch: AP Physics 1 - Problem Solving q+a Simple Harmonic Oscillators