The displacement, velocity and acceleration of the mass are related as shown above. To draw these, think about what the object is doing at each point as it oscillates from the start position described above.
As it passes through the equilibrium position on the way down it's at maximum speed down (negative), its displacement is zero and because the spring is at its equilibrium position, there is no resultant force on the mass so it is not accelerating.
At the bottom, the mass stops momentarily as it changes direction, so velocity is zero. The displacement is a maximum in the negative direction, so the acceleration is a maximum in a positive direction as the spring tries to shorten again.
The important point to note is the phase difference between these three variables...
1. The velocity, v, is zero where there are stationary points at the peaks and troughs of the displacement graph and the velocity is a maximum when the displacement is zero. (Don't forget the gradient of the displacement graph will equal velocity.)
2. The displacement and acceleration graphs are 180 degrees out of phase and therefore look like a mirror image of each other in the time axis. (Don't forget the gradient of the velocity graph will equal acceleration.)
Definition of Simple Harmonic Motion:
All of the above leads us to the formal definition of shm:
A body is undergoing SHMwhen the acceleration on the body is proportional to its displacement, but acts in the opposite direction.
Acceleration a displacement
a α - s
It's also important to note that for SHM, the time period of the oscillations is constant and doesn't change even if the amplitude is changing.
There are two common examples of simple harmonic motion:
SHM is used to explain the behaviour of atoms in a lattice, which oscillate like masses on springs.
Question:
Calculations and Examples with SHM
Finding Acceleration
The definition for simple harmonic motion tells us that:
a α -s
We can get rid of the proportionality sign by putting in a constant. In this case, the constant is (2πf)2, so:
a = - (2 πf)2 s
Example:
A road drill vibrates up and down with SHM at a frequency of 20Hz.
What's the maximum acceleration of the pick head if the amplitude of the oscillation is 5cm?
Answer:
The maximum acceleration occurs at the point of maximum displacement - for instance, the amplitude. So,
a = -(40π)2 x 5 x 10-2
a = 790ms-2
Once you've found the acceleration, you can calculate the forces involved - so long as you are told the mass of the pick head.
So if the mass of the pick head is 3kg...
F = ma = 3 x 790 = 2370N (3sf)
Finding Displacement and Velocity
As shm oscillations follow a sine or cosine wave, we can find the displacement at any point using:
Where:
A = amplitude - not acceleration!
Velocity can be found using:
Example:
A person uses a rope swing to get across a stream. They run and grab the rope at A and swing to the other side at B, 6m from the start.
They hang on to the rope until it stops at the end of the swing at C, 6.5 m from the start.
The centre of the swing (and of the stream), D, is now 3.5 m away, so the amplitude of the swing is 3.5 m.
a) If the frequency of the swing is 0.2 Hz, what was the person's speed as they passed the opposite bank of the stream at B, 3 m from the centre of the swing?
b) What was the maximum speed during the swing and where exactly was the person then?
Kinetic and Potential Energy
The total energy during an oscillation (for example, a pendulum on a string) is constant as long as no energy is lost to the environment:
The energy in the system changes from potential to kinetic and back every half cycle, but the total energy in the system is constant at all times (the dotted line is the sum of the P.E. and K.E.)
Note: That energy changes from KE to PE and back again twice in every cycle.
Total energy = Kinetic energy + potential energy
This applies to a mass oscillating on a spring, so we can easily calculate the total energy using the equations for kinetic energy of a mass and the potential energy stored in a spring.
KE = 1/2 mv2 and PE = 1/2 ks2
So,
Total energy = 1/2 mv2 + 1/2 ks2
Where:
m = mass on the spring (kg)
v = velocity of the mass (ms-1)
k = spring constant
s = displacement of mass (m)
Energy and Amplitude
The amplitude of a wave gives an indication of the amount of energy the oscillator has. This makes sense if you think of the spring and mass. The greater the amplitude the larger the amount of energy stored in the spring when it is extended. However,
PE = 1/2 ks2 so energy must be proportional to the amplitude2.
Damping, Natural Frequency and Resonance
Damping
In practice, the amplitude of vibrations becomes progressively smaller as energy is lost due to friction between the oscillating body and the particles in the air.
If energy is being removed from the system, the amplitude of the oscillations must become smaller and smaller, we say that the oscillations are being damped.
- The amplitude of oscillations decrease with time.
- The higher the damping, the faster the oscillations will reduce in size.
Critical damping is the damping required to make the oscillations stop in the quickest possible time without going past the equilibrium position.
Damping of free vibrations:
Damping of forced Vibrations:
Note: That the lines in the graph never touch or cross. Also, note that if the system becomes heavily damped, the peak of the red line will move slightly to the left - to a slightly lower value of natural frequency.
It is sometimes useful to damp vibrations. For example, car suspensions are damped to stop them bouncing for a long time.
However, if the car suspensions are over damped then the car may jolt uncomfortably every time the car goes over a bump in the road. Over damping also means that there is a long delay before the suspension can react to any more bumps.
Question:
Natural Frequency
Hit anything and it will vibrate. The amazing thing is that every time you hit it, it will vibrate with exactly the same frequency, no matter how hard you hit it.
The frequency of un-damped oscillations in a system, which has been allowed to oscillate on its own, is called the natural frequency, f0.
In order to keep it vibrating after you've hit it, you need to keep re-hitting it periodically to make up for the energy being lost. We say that you need to apply a periodic force to it. (Although some people would just say that you are being unnecessarily violent.)
The frequency with which the periodic force is applied is called the forced frequency. If the forced frequency equals the natural frequency of a system (or a whole number multiple of it) then the amplitude of the oscillations will grow and grow. This effect is known as resonance.
Don't try this at home!
Interesting point: During resonance vibrations can build up to dangerous levels...
- Washing machines and buses will often vibrate violently when the engine oscillates at their natural frequency.
- It is resonance that smashes a glass when an opera singer hits the note that is the natural frequency of the glass.
- Resonance is also why soldiers break their march to cross a bridge - otherwise resonance may cause the bridge to vibrate so violently that it collapses.
Resonance has many uses, for instance:
- Musical instruments - for example, wind and string instruments.
- Circuits can use electrical resonance - for example, for selecting communication channels.
Question:
Match the following words to the correct description:
