# Period of Oscillation of a Simple Pendulum

Andrew Rudhall 11H

Science 1 - Physics

Centre No: 63203                                Candidate No: 7152

Period of Oscillation of a Simple Pendulum

Aim

To find out what factors affect the period of oscillation of a simple pendulum.  I hope to find what these factors are by varying factors such as angle of release, mass of pendulum bob and length of pendulum.

Background Information

• Pendulum

This is a device which consists of an object (pendulum bob) suspended from a fixed point that swings back and forth whilst it is under the influence of gravity.  Due to the constant at which pendulums swing, they are excellent time measuring pieces and eventually led to the invention of the pendulum clock, which was the most accurate time measuring piece at that time.

• Galileo’s Formula

The acceleration due to gravity of an object in freefall was a very important notion for Galileo to realise.  There is a simple law that states the distance travelled in freefall is proportional to the square of the time elapsed.

It was during a Sunday mass in Pisa when he noticed a chandelier above him swinging to and fro.  Using his pulse as a clock, Galileo saw that the period of the swing was independent of how far it swung.  Only the length of the pendulum made any difference to the time required for a swing.

This formula allows the period of a pendulum to be calculated:

P = 2∏√(l/g)

• P is the period of oscillation of the pendulum (in seconds).
• L is the length of the pendulum (in metres).
• G is the acceleration of gravity (on Earth this is counted as 9.8).
•  has the value of 3.14159...etc.

Prediction and Hypothesis

Length is a certain factor, which will determine the time of oscillation of a pendulum.  It is difficult to state why this is, but it helps to actually visualise a pendulum swinging.  The shorter the pendulum is, the faster it will oscillate and visa versa - the longer the pendulum is, the longer it will take for each oscillation to take place.  This is because of the effect gravity has on the pendulum.  In a short pendulum, it must swing quickly because it would defy gravity otherwise.  As it is difficult to explain this concept, I shall draw a diagram, which may help to explain the situation.

As you can see, pendulum ‘b’ has a further distance to travel than pendulum ‘a‘.  Mathematically, it is possible to work out the distances using the following formula:

d (diameter) ÷ (360° ÷ x°)

If this formula is applied to both pendulums, distance ‘a’ equates to 1.57 cm and distance ‘b’ equates to 3.14 cm (double the distance of distance ‘a’).  If mass does not affect gravitational acceleration (which I will discuss later on in the hypothesis), then the pendulums will swing at the same speed.  Therefore, the pendulum should take twice as long to complete one oscillation where the length is double.  Consequently, time of oscillation is certainly affected by the length of the pendulum.

Energy is converted into heat in order to overcome air resistance.  Eventually, the pendulum will come to a halt.  In theory, a constant swinging motion could only be achieved inside a vacuum, and one major source of gravity.  If no air resistance can act on the pendulum, then none of the energy from the swinging motion is converted into heat due to friction as the pendulum moves. Therefore it is air resistance and pivot resistance that pendulums lose their energy from leading to the loss of height gained each time an oscillation is completed.  However, pendulums have to overcome the air resistance, so my experiments will not be completely accurate.

The mass of an object should not affect the rate of oscillation if the air resistance is equal.  For example, two cannon balls that are dropped from the same height, but both have different masses, in theory; they should hit the ground at exactly the same time.  It is only air resistance, which prevents objects from falling at a uniform speed.  Another example of this would be a pebble and a feather at the same mass.  Obviously, the feather would be considerably slowed if there were air resistance acting on it.  However in a vacuum, both the pebble and the feather would fall at exactly the same speed, as would any other object.  As the rate of oscillation is linked to gravity, if I had two pendulum bobs, one made out of polystyrene and one out of lead, both of identical in size and shape, there would be absolutely no difference in the rate of oscillation.  Therefore, I predict that both density and mass will have absolutely no effect on the oscillation of a pendulum.

Below is a diagram showing the forces involved when a simple pendulum swings.

In theory, both pressure and temperature will have a minute effect on the rate of oscillation.  This is because the higher the pressure, the more air molecules there are to act on the pendulum.  The bob will need to exert a greater level of force to overcome the resistance of the air.  Temperature will also affect the density of the air particles.  The hotter it is, the less dense the air will be, and therefore meaning there will be fewer air particles to overcome.  The hotter the temperature it is, the more efficient the pendulum is.  However, both of these variables are very small and will not a very big effect on the experiment.

It is difficult to say whether angle of release will have any effect on the rate of oscillation or not.  If the pendulum bob is released at an angle of 45° instead of 25°, then there will be a difference in speed.  If the length is kept the same, then in theory, each oscillation should be identical in time.  But due to gravitational acceleration, the larger the angle, the faster the pendulum will go.  As speed increases, so does the amount of air resistance.  For example if one swims, it is difficult to go very fast as there is ...