Analyse the motion of an oscillating body.
Define quantities used to describe oscillations.
Understand the sinusoidal nature of oscillations.
The simple pendulum
There are many oscillating systems we could investigate but we will start with the simple pendulum. Simple means:
- the mass is very small (ideally a point)
- the string is inelastic (doesn’t stretch)
- the swings are small (< 10°)
One complete swing from A – B – A
The position where the pendulum bob would come to rest.
The maximum displacement from the equilibrium position.
Time period (T)
The time for one cycle
The number of cycles per second f = 1/T (unit hertz)
In this practical a video of the motion of an oscillating object will be analysed using LoggerPro to show that its motion is sinusoidal.
To record the video you can use your webcam, camera or mobile telephone. The video will be analysed with LoggerPro so you should save the video in a format that can be played with quicktime (mov, avi, mpeg or wav). If using your webcam the movie can be recorded directly into LoggerPro, this is maybe the simplest way.
Set up a pendulum in front of your camera so that the oscillation stays within the video frame. Before starting to record find an object of known length (a ruler or piece of card) that also fits in the frame when placed the same distance from the camera as the pendulum, this will be used to scale the video.
- Start recording then place the scaling object into the field of view inline with the pendulum then remove it.
- Start the pendulum swinging and record about 5 swings.
- measure the length of the pendulum from support to centre of mass.
If you are recording with a camera you will need to insert the video into LoggerPro
If you remember how to use video analysis in LoggerPro then track the position of the pendulum as it swings and plot the x displacement against time.
- Use your scaling object to set the scale
- Set the origin to be the equilibrium position
- Track the motion for one complete cycle (there and back).
If you can't remember here is a video.
- Plot the best fit sine curve for you data.
- The curve will be of the form x = A sin (B*t + C) + D, what do A, B and C represent. You can try changing them to see what happens.
- The time period of a pendulum is given by the formula
See if this is consistent with your results.
- Plot X velocity and X displacement on the same graph. Note the different phase.
- Why isn't the velocity data such a smooth curve?
Sinusoidal nature of pendulum motion
From the video analysis you have seen that the equation for the motion is sinusoidal.
- Try varying the amplitude and frequency to see the effect on the graph.
This line has the equation:
- What is the value of x at time t = 0?
- What is the value of x after 1 time period (t = T)?
- In which direction is the pendulum bob travelling at time t = 0?
Analysing the forces
To understand why the motion is sinusoidal we need to have a look at the forces.
Before we start it is important to remember that this is a simple pendulum so the swing is very small, so small that we can assume the motion is horizontal, this means that the vertical forces are balanced.
- Write an equation for the vertical component of T.
- If the angle is small what does cosθ approximately equal?
- Show that for small angles
T = mg
- If the swing is large what is the direction of the resultant force at the bottom of the swing?
The reason that the pendulum swings down is because there is a force acting towards the equilibrium position this is the horizontal component of T.
- Write an equation for the horizontal component of T.
- Write an equation for the sinθ in terms of L and x.
- Substitute for sinθ
According to Newton's second law this unbalanced force will equal ma where a is the acceleration of the bob towards the equilibrium position.
- Show that this gives
If you now substitute for T = mg you get:
Notice that I have put in a negative sign. This is because in 1D the acceleration is in the opposite direction to the displacement.
This means that the acceleration is directly proportional to the displacement but acts in the other direction, this type of motion is called simple harmonic motion.
Simple harmonic motion
Simple harmonic motion is defined as motion where the acceleration is directly proportional to the displacement from a fixed point and always acts towards that point.
You can see how this leads to an oscillation by building an iterative model in excel : Iterative SHM (Excel)
If the displacement of a particle is sinusoidal then its acceleration will also be sinusoidal. This can be demonstrated in Algodoo.
- Make a pendulum in Algodoo. If the string is too thick then zoom out and start again. To draw a straight string press shift.
- Plot the horizontal displacement, velocity and acceleration for the ball. Remember, if you drag the graphs to one side they will stay in view.
- Notice that the acceleration changes in time with the displacement but with opposite sign.
The velocity time graph is the gradient of the displacement time and the acceleration time is the gradient of the velocity time. You can see how this works with the phET calculus grapher simulation below. The derivative is the gradient. You can see how the derivative of a sin curve is a cos curve and the derivative of cos is -sin.
Using circular motion to derive the equations of SHM (HL)
We know that acceleration is proportional to displacement but what is the constant of proportionality? We can find this by differentiating x = sin2πft twice but if you haven't done this in maths it won't make sense. Alternatively we can use what we know about circular motion.
Consider a body moving in a circle with constant speed v and radius r. At time t the body is in the position shown.
- Write an expression for the horizontal displacement in terms of θ.
- If the angular velocity is ω show that
Now lets consider the acceleration. We know that this is directed towards the centre and equals ω2r (centripetal acceleration).
- Write an expression for the horizontal component of the acceleration and show that
we can now see that
But since a and x are vectors and have opposite direction we should write
So the horizontal motion is SHM we can see this in GeoGebra
Note that in SHM ω is called the angular frequency (2πf) this is equivalent to angular velocity but there isn't any rotation happening.
This method can also be used to derive an equation for the velocity of the particle at any position.
- Write an equation for the horizontal component of the velocity.
You will probably know that Pythagoras' theorem can be written
Use this expression to show that
Cos or sin?
In the circular motion example the horizontal motion was r cosωt but in the pendulum example we had r sinωt. This doesn't matter since the shape of the graph is the same, its just the starting point that's different.
- If the displacement of a pendulum is r cosωt where did it start?
- If displacement = -r sinωt where was the pendulum at t = 0 and in which direction was it travelling?
Summary of equations
Exercises 1 - 9 chapter 5
Time period of a pendulum
Earlier in this activity you showed that the acceleration of a pendulum bob is given by the equation:
Now you know that the general equation for all SHM is
Equate these two equations and show that for a simple pendulum
So the pendulum can be used to find the acceleration due to gravity, here is the worksheet Measuring g with a pendulum