Simple harmonic oscillations (5.3.1)
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Displacement is the distance an object is from the equilibrium position (the equilibrium is the point where an oscillating object will come to rest). Displacement is represented by in equations and has the SI unit of metres ().
Amplitude is the maximum distance an object can be from the equilibrium. Amplitude is represented by in equations and has the SI unit of metres ().
Period or time period is the time taken for an oscillating object to complete one full oscillation. Period is represented by and has the SI units of seconds ().
The diagram below shows the path of an ideal pendulum over one complete period. The pendulum starts at the right amplitude, swings past the equilibrium point to the left amplitude, and then returns to its starting point.
Frequency is the number of full oscillations completed per second. Frequency is represented by and has SI units of hertz (). It can be calculated from the time period using the following formula:
For the calculated frequency value to be in hertz, the time period value must be in seconds.
Angular frequency is defined as how quickly an object completes one full oscillation (one time period). The shorter the time period of the oscillation, the greater the angular frequency.
Angular frequency is represented by in equations and has SI units of radians per second
Phase difference measures how much a point on a wave is ahead or behind a point on either the same wave or a different wave.
The SI units for phase difference are either radians or degrees or fractions of a wavelength, One complete wave is defined as being equal to or radians. This is because one full cycle of periodic motion is equivalent to a complete rotation around a circle.
In the example below, the trough of the red wave is ahead of a matching trough on the blue wave.
A whole wave is so a phase difference of is also equal to
A whole wave is also equal to radians, so a phase difference of is also equal to
Question walkthrough
Phase Difference Between Two Wave Points
Finds a microwave's wavelength from its frequency, then calculates the phase difference in degrees between two points a given distance apart.
Angular frequency is defined as how quickly an object completes one full oscillation (one time period). The shorter the time period of the oscillation the greater the angular frequency.
It is represented by and has SI units of radians per second
Whereas linear speed measures distance travelled per unit of time, angular frequency measures the angle subtended per unit of time.
It can be found using the equations below:
It can be calculated from either an object’s period, or frequency,
In order for the calculated angular frequency to be in SI units of radians per second the time period value must be in seconds and the frequency value must be in hertz.
A simple pendulum and a mass on a spring are examples of simple harmonic oscillators (SHO). These objects oscillate with what is known as simple harmonic motion (SHM).
Simple harmonic motion occurs when an object oscillates to and fro, either side of an equilibrium position. The equilibrium is the position in which it a real oscillator would come to rest due to friction.
A restoring force tries to return the oscillator to equilibrium. The restoring force could be gravity, tension, or other forces. As shown in the diagram below, the size of the restoring force is directly proportional to the size of the displacement from equilibrium.
Newton’s second law states that the size of the force is directly proportional to the size of the acceleration. Therefore, when the restoring force towards the equilibrium is greatest, so is the acceleration towards the equilibrium, as shown in the diagram below.
The definition of an object moving with simple harmonic motion (SHM) has two key conditions:
An oscillation in which the acceleration of an object is directly proportional to its displacement from its equilibrium position, and is directed towards the equilibrium, expressed mathematically as:
Where:
- represents the acceleration measured in metres per second
- is the proportionality constant.
- is displacement from the equilibrium measured in metres
The negative sign represents that the acceleration is in the opposite direction to the displacement, as it is always directed towards the equilibrium.
In a proportional relationship, there is always a constant . In the SHM equation, the full equation is as follows:
The constant in this case is actually the angular frequency squared, and is measured in
Techniques and procedures exist to measure the time period of an object moving with simple harmonic motion. For this example, we will look at an experiment to determine the time period of a simple pendulum.
Set up the experiment shown in the diagram below. A fiducial marker is used to mark when to start and stop timings as the pendulum passes it. It should be placed at the equilibrium position. The marker can be something simple, like a pin stuck in a cork.
For finding the time period of a mass on a spring, you would use a similar setup with a mass being suspended from a spring. In this case, the fiducial marker should be placed horizontally at the equilibrium position, which is where the mass will be at rest when it is not oscillating.
Determining the frequency of a real object moving with simple harmonic motion requires measuring the time period.
Using the fiducial marker placed at the equilibrium as your reference for starting and stopping the stopwatch, one complete time period would look like the diagram below:
One time period is often short in practice, so to reduce uncertainty, you should measure at least ten time periods. Then divide the measurement of ten time periods by ten to find the value of one time period.
Once you have the time period, you can find the frequency using the formula below, making sure to enter the time period in seconds:
There are two equations you can use to calculate the displacement of an object at any time that is moving with simple harmonic motion.
The first equation or second equation below should be used if the oscillations begin at maximum displacent or the equilibrium positions respectively.
Where:
- is the object’s displacement from the equilibrium position. It is measured in metres.
- is the amplitude, which is the maximum displacement of the object from equilibrium, measured in metres.
- is the angular frequency, defined as how quickly an object completes one full oscillation (one time period).
- is the time since the oscillations started, measured in seconds.
Displacement and amplitude are vectors, so can be positive or negative. Moreover, the shorter the time period of the oscillation, the greater the angular frequency. The angular frequency remains constant throughout simple harmonic motion as the time period is constant.
It is important to note that your calculator should be in radians mode when using these equations.
In order to find the acceleration of an object at a certain displacement from the equilibrium, we can use the equation which is defined by the definition of simple harmonic motion.
In this equation,
- is the angular frequency and is measured in
- is acceleration measured in
- is the displacement from the equilibrium position measured in metres.
Question walkthrough
Finding SHM Displacement at Given Time
Uses the SHM displacement equation x=Acos(ωt) to find a pendulum's displacement at a specific time, given its amplitude and frequency.
Question walkthrough
SHM Acceleration from Displacement and Period
Finds the time period from an oscillation count, then uses the SHM acceleration equation to find a pendulum's acceleration at a given displacement.
The velocity of an object moving with simple harmonic motion at a particular displacement from the equilibrium can be calculated using this equation:
In this equation:
- is velocity measured in metres per second
- is the angular frequency and is measured in radians per second
- is the amplitude measured in metres
- is the displacement from the equilibrium measured in metres
The reason for the is that an object can pass through the equilibrium from both directions with positive or negative velocity.
An object moving with simple harmonic motion has its maximum velocity at the equilibrium, when :
Therefore, the equation for the velocity of an object moving with SHM can be simplified to this:
Question walkthrough
Finding Speed in Mass-Spring SHM
Uses the SHM speed equation to find a mass-spring system's speed at a given displacement and at equilibrium, from its amplitude and time period.
An oscillator moving with simple harmonic motion is known as an isochronous oscillator. Isochronous oscillations mean that the time it takes to complete one oscillation is independent of its amplitude, for example, no matter how much you initially displace a pendulum or a mass on a spring; its time period remains the same.
The time period of a simple pendulum in seconds can be calculated using the equation below:
The equation shows that the only variables that can change the time period are:
- the length of the pendulum which is measured from the point of suspension to the centre of mass of the bob, the units will be metres and
- the acceleration due to gravity measured in metres per second per second
The time period of a mass on a spring in seconds can be calculated using the equation below:
The only variables that can change the time period of an oscillation are:
- the mass, measured in kilograms and
- the spring constant, measured in newtons per metre
Question walkthrough
Finding Pendulum Length from Time Period
Rearranges the simple pendulum time period formula to find its length, testing that the period is independent of amplitude for SHM.
A pendulum moves with simple harmonic motion. The graph below shows how the pendulum’s displacement would be represented graphically over one time period, with the pendulum initially displaced to the left of the equilibrium with positive amplitude.
- Initially the pendulum bob starts at the positive amplitude.
- Quarter of a time period, the pendulum bob passes through the equilibrium, so the displacement equals zero.
- Half a time period the pendulum bob is now at the negative amplitude
- Three-quarters of a time period, the pendulum bob is passing back through the equilibrium in the opposite direction.
- Full time period the pendulum bob is now back at its starting position, which is the positive amplitude.
A pendulum moves with simple harmonic motion. The graph below shows how the velocity of the pendulum would be represented graphically over one time period. Starting with the pendulum displaced to the left of the equilibrium with positive amplitude.
- Initially the pendulum bob starts at the positive amplitude. At either amplitude, the velocity of the bob is momentarily zero.
- Quarter of a time period the bob passes through equilibrium where the velocity is maximum. The bob moves towards the negative amplitude with negative velocity.
- Half a time period the bob is at the negative amplitude. The velocity is momentarily zero.
- Three-quarters of a time period the bob moves through the equilibrium in the opposite direction. It has maximum positive velocity.
- Full time period the bob returns to its starting position; both the positive amplitude and velocity are zero.
A pendulum moves with simple harmonic motion. The graph shows a graphical representation of the acceleration of a pendulum over one time period. We begin with the pendulum displaced to the left (which we are indicating as the positive amplitude).
- Initially the restoring force is towards the right (negative side). Force and acceleration are proportional to displacement, so acceleration is greatest at the amplitude.
- Quarter of a time period the pendulum bob passes through equilibrium. Acceleration is proportional to displacement from the equilibrium, so at equilibrium, acceleration equals zero.
- Half a time period the bob is now at the negative amplitude. The restoring force and acceleration are greatest and directed towards the positive side.
- Three-quarters of a time period the bob passes through the equilibrium in the opposite direction, so acceleration is zero again.
- Full time period the bob returns to its starting position and the positive amplitude and acceleration are greatest in the negative direction.
The diagram below shows a pendulum undergoing simple harmonic motion. The displacement, velocity and acceleration of the bob are shown together over a full time period, so you can observe its motion and analyse the phase difference between them graphically.
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The definition of simple harmonic motion states that the acceleration of an object moving with simple harmonic motion is directly proportional to the displacement from the equilibrium and is always directed towards the equilibrium position.
The graph shows how an object’s acceleration varies with displacement; it is represented by a straight line that passes through the origin.
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- When the object is at its maximum negative amplitude, it will experience the greatest positive acceleration.
- When at the equilibrium, it will experience no acceleration.
- When at the maximum positive amplitude, it will experience the greatest negative acceleration.
An object moving with simple harmonic motion has the greatest velocity when it passes through the equilibrium position. However, the sign of the velocity changes depending on the direction of travel.
When the object is at either maximum amplitude, its velocity is zero, representing the moment it changes direction.
Shown graphically, these four points are joined by a circle.
