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6.2 Energy of a Simple Harmonic Oscillator

8 min readapril 13, 2023

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

Conservation of Energy

The energy of a system is conserved.

Internal energy

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

Here are some key things to know about the of a :

  • The of an object is the energy associated with the random, chaotic motion of its constituent particles. It is a measure of the of the object and is often symbolized by the letter U.

  • In a , the is stored in the form of when the oscillator is displaced from its . As the oscillator oscillates back and forth, the of the system is continually converted into and back into .

  • The of a is periodic, meaning that it follows a repeating pattern over time. The of the oscillator is at a maximum when the oscillator is at its maximum from its and is at a minimum when the oscillator is at its .

  • The of a can be calculated using the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

  • The of a 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 over time.

Potential Energy

A system with internal structure can have . exists within a system if the objects within that system interact with .

Here are some key things to know about the of a :

  • The of an object is the energy that an object possesses due to its position or configuration within a . It is a measure of the potential for the object to do and is often symbolized by the letter U.

  • In a , the of the system is stored in the form of when the oscillator is displaced from its . 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 .

  • The of a is periodic, meaning that it follows a repeating pattern over time. The of the oscillator is at a maximum when the oscillator is at its maximum from its , and is at a minimum when the oscillator is at its .

  • The of a can be calculated using the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

  • The of a 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 over time.

Kinetic Energy

The of a system includes the of the objects that make up the system and the of the configuration of objects that make up the system.

Here are some key things to know about the of a :

  • The 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 due to its motion and is often symbolized by the letter K.

  • In a , the of the system is stored in the form of when the oscillator is moving. This energy is due to the motion of the oscillator as it oscillates back and forth.

  • The of a is periodic, meaning that it follows a repeating pattern over time. The of the oscillator is at a maximum when the oscillator is at its maximum , and is at a minimum when the oscillator is at its or at a point of maximum from the .

  • The of a can be calculated using the equation: K = 1/2*mv^2, where K is the , m is the mass of the oscillator, and v is the of the oscillator.

  • The of a 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 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 occurs when the spring is stretched (or compressed) the most, and the maximum occurs at the equilibrium point. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fo.PNG?alt=media&token=09fe7d2d-56c7-4a21-9e15-d7e12dc11a4e

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 in the spring. In pictures B and D, the mass is at the (x=0) and all the energy is now .

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fgr.PNG?alt=media&token=64eeb3f0-13d0-4243-8395-c9b3846942e1

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 on the spring-mass system

  2. The and graphs are curves. Because of the squared term in the equation, we expect this. If the term is to the 1st power, then the graph would be linear.

  3. The is greatest when the position graph is at its maximum. The is greatest when the graph is at its maximum.

Example Problem 1:

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

Solution:

The total energy of a is the sum of its and .

The of a is given by the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

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

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

At the maximum from the , the of the oscillator is zero and the is at a maximum.

Therefore, the total energy of the oscillator at the maximum from the 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 from the is 1 J.

Example Problem 2:

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

Solution:

The total energy of a is the sum of its and .

The of a is given by the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

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

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

At the , the of the oscillator is zero and the is at a minimum.

Therefore, the total energy of the oscillator at the 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 is 12.5 J.

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

Key Terms to Review (16)

Conservation of Energy

: The principle that states that energy cannot be created or destroyed but can only be transferred or transformed from one form to another.

Conservative Forces

: Conservative forces are forces that do not dissipate mechanical energy as they act on objects within a system. They conserve mechanical energy by transforming one form of mechanical energy into another without any loss.

Displacement

: Displacement refers to the change in position of an object from its initial point to its final point, taking into account both distance and direction.

Elastic Potential Energy

: Elastic potential energy is the stored energy in an elastic material when it is stretched or compressed. It depends on the amount of deformation and the spring constant of the material.

Equilibrium position

: The equilibrium position is the stable, balanced point where an object or system experiences no net force and remains at rest or in uniform motion.

Force field

: A force field is a region in space where an object experiences a force due to the presence of another object or objects.

Harmonic motion

: Harmonic motion refers to repetitive back-and-forth motion around an equilibrium position where acceleration is proportional to displacement and directed towards it.

Internal Energy

: Internal energy refers to the total amount of kinetic and potential energies possessed by all particles within a substance.

Kinetic Energy

: Kinetic energy is the energy an object possesses due to its motion. It depends on both the mass and velocity of the object.

Periodic motion

: Periodic motion refers to the repetitive back-and-forth movement of an object or system in a regular pattern over time.

Potential Energy

: Potential energy is the stored energy an object possesses due to its position or condition. It can be converted into other forms of energy, such as kinetic energy, when released.

Simple Harmonic Oscillator

: A simple harmonic oscillator refers to any system that exhibits periodic motion back and forth around an equilibrium position under the influence of a restoring force proportional to its displacement.

Spring Constant

: The spring constant represents how stiff or flexible a spring is. It determines how much force will be required to stretch or compress a spring by a certain distance.

Thermal Energy

: Thermal energy refers to the internal energy of an object due to the motion and vibration of its particles.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It includes both speed and direction.

Work

: Work is the transfer of energy that occurs when a force is applied to an object and it moves in the direction of the force. It is equal to the change in kinetic energy of the object.

6.2 Energy of a Simple Harmonic Oscillator

8 min readapril 13, 2023

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

Kanya Shah

Kanya Shah

Daniella Garcia-Loos

Daniella Garcia-Loos

Conservation of Energy

The energy of a system is conserved.

Internal energy

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

Here are some key things to know about the of a :

  • The of an object is the energy associated with the random, chaotic motion of its constituent particles. It is a measure of the of the object and is often symbolized by the letter U.

  • In a , the is stored in the form of when the oscillator is displaced from its . As the oscillator oscillates back and forth, the of the system is continually converted into and back into .

  • The of a is periodic, meaning that it follows a repeating pattern over time. The of the oscillator is at a maximum when the oscillator is at its maximum from its and is at a minimum when the oscillator is at its .

  • The of a can be calculated using the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

  • The of a 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 over time.

Potential Energy

A system with internal structure can have . exists within a system if the objects within that system interact with .

Here are some key things to know about the of a :

  • The of an object is the energy that an object possesses due to its position or configuration within a . It is a measure of the potential for the object to do and is often symbolized by the letter U.

  • In a , the of the system is stored in the form of when the oscillator is displaced from its . 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 .

  • The of a is periodic, meaning that it follows a repeating pattern over time. The of the oscillator is at a maximum when the oscillator is at its maximum from its , and is at a minimum when the oscillator is at its .

  • The of a can be calculated using the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

  • The of a 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 over time.

Kinetic Energy

The of a system includes the of the objects that make up the system and the of the configuration of objects that make up the system.

Here are some key things to know about the of a :

  • The 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 due to its motion and is often symbolized by the letter K.

  • In a , the of the system is stored in the form of when the oscillator is moving. This energy is due to the motion of the oscillator as it oscillates back and forth.

  • The of a is periodic, meaning that it follows a repeating pattern over time. The of the oscillator is at a maximum when the oscillator is at its maximum , and is at a minimum when the oscillator is at its or at a point of maximum from the .

  • The of a can be calculated using the equation: K = 1/2*mv^2, where K is the , m is the mass of the oscillator, and v is the of the oscillator.

  • The of a 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 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 occurs when the spring is stretched (or compressed) the most, and the maximum occurs at the equilibrium point. 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fo.PNG?alt=media&token=09fe7d2d-56c7-4a21-9e15-d7e12dc11a4e

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 in the spring. In pictures B and D, the mass is at the (x=0) and all the energy is now .

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fgr.PNG?alt=media&token=64eeb3f0-13d0-4243-8395-c9b3846942e1

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 on the spring-mass system

  2. The and graphs are curves. Because of the squared term in the equation, we expect this. If the term is to the 1st power, then the graph would be linear.

  3. The is greatest when the position graph is at its maximum. The is greatest when the graph is at its maximum.

Example Problem 1:

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

Solution:

The total energy of a is the sum of its and .

The of a is given by the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

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

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

At the maximum from the , the of the oscillator is zero and the is at a maximum.

Therefore, the total energy of the oscillator at the maximum from the 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 from the is 1 J.

Example Problem 2:

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

Solution:

The total energy of a is the sum of its and .

The of a is given by the equation: U = 1/2*kx^2, where U is the , k is the , and x is the of the oscillator from its .

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

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

At the , the of the oscillator is zero and the is at a minimum.

Therefore, the total energy of the oscillator at the 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 is 12.5 J.

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

Key Terms to Review (16)

Conservation of Energy

: The principle that states that energy cannot be created or destroyed but can only be transferred or transformed from one form to another.

Conservative Forces

: Conservative forces are forces that do not dissipate mechanical energy as they act on objects within a system. They conserve mechanical energy by transforming one form of mechanical energy into another without any loss.

Displacement

: Displacement refers to the change in position of an object from its initial point to its final point, taking into account both distance and direction.

Elastic Potential Energy

: Elastic potential energy is the stored energy in an elastic material when it is stretched or compressed. It depends on the amount of deformation and the spring constant of the material.

Equilibrium position

: The equilibrium position is the stable, balanced point where an object or system experiences no net force and remains at rest or in uniform motion.

Force field

: A force field is a region in space where an object experiences a force due to the presence of another object or objects.

Harmonic motion

: Harmonic motion refers to repetitive back-and-forth motion around an equilibrium position where acceleration is proportional to displacement and directed towards it.

Internal Energy

: Internal energy refers to the total amount of kinetic and potential energies possessed by all particles within a substance.

Kinetic Energy

: Kinetic energy is the energy an object possesses due to its motion. It depends on both the mass and velocity of the object.

Periodic motion

: Periodic motion refers to the repetitive back-and-forth movement of an object or system in a regular pattern over time.

Potential Energy

: Potential energy is the stored energy an object possesses due to its position or condition. It can be converted into other forms of energy, such as kinetic energy, when released.

Simple Harmonic Oscillator

: A simple harmonic oscillator refers to any system that exhibits periodic motion back and forth around an equilibrium position under the influence of a restoring force proportional to its displacement.

Spring Constant

: The spring constant represents how stiff or flexible a spring is. It determines how much force will be required to stretch or compress a spring by a certain distance.

Thermal Energy

: Thermal energy refers to the internal energy of an object due to the motion and vibration of its particles.

Velocity

: Velocity refers to the rate at which an object changes its position in a specific direction. It includes both speed and direction.

Work

: Work is the transfer of energy that occurs when a force is applied to an object and it moves in the direction of the force. It is equal to the change in kinetic energy of the object.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.