Simple harmonic motion is all around us. The periodic motion happens because as an object oscillates the restoring force is proportional to the displacement and in the opposite direction.

Displacement, Amplitude, Frequency and Period

Simple harmonic motion has various terms that you must understand right from the start. Displacement is the distance away from the equilibrium position and can be positive or negative, whereas the amplitude is the magnitude of the maximum displacement. Frequency and period are the same as for waves but used in a different context.

Angular Frequency and Phase Difference

When we look at circular motion it is useful to consider the angular velocity of an object. When we look at simple harmonic oscillators we use the term ‘Angular Frequency’ which although it has the same Greek symbol omega and the same units, is a different quantity. 

Phase difference, with the Greek symbol phi, is also useful when we consider two oscillating objects or the motion of one object at different times.

The Defining Equation for SHM

This is a really important equation to recognise and there are so many examples in everyday life of objects that oscillate with simple harmonic motion. This video shows an example of a simple pendulum and why the motion must be simple harmonic.

Simple harmonic motion is defined as when: the acceleration of the object is directly proportional to its displacement and acts in the opposite direction: a = -kx.

How to Time an Object moving with SHM

This short video shows how you should time objects that move with simple harmonic motion – where the best place to start and stop the stopwatch is and also why timing for more than one cycle is essential.

Solutions to the SHM Equation: x=A cos wt and x=A sin wt

If we look at the motion of an object in simple harmonic motion we can find an equation to describe its sinusoidal motion.

 

There are two equations we can use – either using a sine or a cosine function depending on which point we take as the initial position at time zero. This video shows how they can be differentiated a couple of times to show that they do fit the simple harmonic motion defining equation.

Velocity of an Object Moving with SHM

This is a rather fun derivation if you like such things: however, you only really need to be able to use the final equation to find the velocity of an oscillating object at various points in its cycle or just its maximum velocity. Don’t be scared by the maths - but for many of you watching this it is good to see why your A Level Maths is important. 

Graphs of Displacement, Velocity and Acceleration

Visualising the motion of an oscillator is a really useful and important skill you can learn. By considering the displacement-time, velocity-time and acceleration-time graphs you can investigate the motion of a simple harmonic oscillator.

Deriving the Equation for a Simple Pendulum

Simple pendulums swing back and forth through a small angle and their time period is independent of the mass on the end or the amplitude. That’s why grandfather clocks use a pendulum to keep time, even as the pendulum swung less and less due to friction the time for each oscillation was constant.

Deriving the Equation for a Spring-Mass System

A simple mass-spring system that oscillates can be used to model many other systems in real life. The time period really only depends on two factors – the mass of the oscillator and the stiffness of the spring. Here I show you how to derive this equation.

Energy in a Pendulum or Horizontal Spring-Mass system

You can often solve problems with oscillators by considering the energy transfer from kinetic energy to potential – either gravitational or elastic. You can also show this on a graph to see how the energies change during the cycle.

Energy in a Vertical Spring-Mass System

Things get slightly more complicated when there is energy transferred between three types: kinetic, gravitational and elastic potential energy. This video looks at how they are exchanged in a mass-spring system that is oscillating vertically.

Resonance

Why do buses suddenly start vibrating when waiting at traffic lights? How does a radio pick up a signal and why does holding your car keys near your head boost the signal and allow you to open your car from further away? It’s all due to resonance: where the driving frequency is equal to the natural frequency of an object and causes an objects amplitude to increase massively.

Damping

Most of the time in real life an objects oscillations will die away as air resistance slows things down: until after a few minutes a pendulum stops swinging or a mass on a spring remains at its equilibrium position. This is due to damping where energy is lost from the system. There are several types of damping including light, heavy and critical damping.

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