Energy of a simple harmonic oscillator (5.3.2)

The kinetic energy of an object is given as:

where:

• is the kinetic energy measured in joules ()
• is the mass measured in kilograms ()
• is the velocity measured in metres per second ().

In simple harmonic motion:

• At the equilibrium position, the speed of the oscillator is maximum and, therefore, this is where the kinetic energy is also greatest.
• At the maximum displacement (equal to the amplitude), the velocity is zero; therefore, so is the kinetic energy.

The graph below illustrates the variation in kinetic energy of a pendulum bob over two distinct time periods.

The maximum kinetic energy of the system can be found using the equation:

The maximum velocity of an object moving with simple harmonic motion in is:

Where:

• is the angular velocity measured in radians per second ()
• is the amplitude measured in meters ().

The potential energy of an object in simple harmonic motion depends on the displacement from the equilibrium position:

• At equilibrium, the potential energy of the oscillator is zero.
• At maximum displacement, the potential energy is maximum.

The graph below illustrates the variation in potential energy of the oscillator over two time periods

If the potential energy of the oscillator is in the form of gravitational potential energy, then we can use the following equation:

where:

• is the height above the equilibrium and is measured in metres ()
• is the mass measured in kilograms ()
• is acceleration due to gravity measured in metres per second squared ().

If the potential energy is in the form of elastic potential energy, then we can use the following equation:

where:

• is the stiffness constant of the object and is measured in Newtons per metre ()
• is the change in length of the object and is measured in metres ().

The maximum elastic potential energy will be when the change in length is equal to the maximum displacement (amplitude), so the equation becomes:

where:

• is the amplitude measured in meters ().

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

The total energy remains constant unless an external force, such as friction, acts on the system which causes a damping effect.

An example of this is a pendulum bob undergoing simple harmonic motion:

1. At the top of the swing, the bob is stationary and all of its energy is in the form of potential energy.
2. As the bob descends, it gains speed and hence kinetic energy, but loses potential energy as its height decreases.
3. At the equilibrium position, all of the bob’s energy is in the form of kinetic energy.
4. As the bob rises again, it loses speed and, hence, kinetic energy, but gains potential energy again.

To calculate the total energy of the system, it is easier to calculate the maximum value of the kinetic energy or the potential energy than the sum of the kinetic energy and potential energy at some point between the equilibrium and the amplitude:

As an object moving with simple harmonic motion oscillates between two amplitudes, it constantly exchanges energy between kinetic and potential forms.

• The kinetic energy is greatest at the equilibrium, where the object is moving fastest and zero at the amplitude.

• The potential energy is greatest at the amplitudes and zero at the equilibrium.

• The total energy is the sum of the kinetic and potential energies at any particular point and is a constant as long as no external forces are acting, such as friction.

Shown below is a graph demonstrating how the different types of energy vary with displacement from the equilibrium position.

Energy changes for an ideal simple pendulum over time, where is the displacement from the equilibrium position:

Energy changes for an ideal mass spring system over time, where is the displacement from the equilibrium position: