The determination of the acceleration due to gravity at the surface of the earth, g, using a simple pendulum.

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Mehreen Tahir

AS Physics Coursework

The Determination of the acceleration due to gravity at the surface of the earth, g, using a simple pendulum.

Aim: The determination of the acceleration due to gravity at the surface of the earth, g, using a simple pendulum

Background

All bodies exert gravitational forces on one another. A large mass, such as the earth produces a gravitational field.

Gravity acts between any two objects, causing a force of attraction which tries to pull the two objects together. An example is if you release an object above the ground it falls down as it is pulled down by gravity.

My investigation aims to allocate a value to the acceleration this object would experience due to the earth’s gravity.

Isaac Newton first discovered gravity when an apple fell on his head. He discovered that every object has a mass and that two masses attract each other. Newton wanted to calculate the gravitational field strength of the earth i.e. what size the acceleration objects, such as the apple experienced, due to the earth’s gravity.

I am going to determine this value, not using a free-fall object, but a simple pendulum. A simple pendulum consists of a mass, suspended from a fixed point by a length of string. The forces acting on the bob (mass) of the pendulum are tension in the string (t) and weight (mg) as shown.

When the bob is displaced from its equilibrium position, with the thread taut and then released it oscillates about the equilibrium position in a fixed vertical plane. An object suspended in the gravitational field of the earth hangs in its equilibrium position when its centre of mass is directly below the fixed point of suspension.

If the maximum displacement of the string from the vertical equilibrium is less than 15˚, the pendulum is found to execute simple harmonic oscillations along the arc of a circle as shown. Simple harmonic motion is defined as oscillation motion of an object about a fixed point.

The time (in seconds) the bob takes to pass once, back and forth over this arc, is called its period. (T)

The period (T) of a pendulum can be related to the acceleration due to gravity and the length of the pendulum (L) by:

T represents period (in seconds)

L is the length of the string from the fixed point to the middle of the bob (in meters)

g is acceleration due to gravity

This tells us that, assuming g is constant at a given place on the earth, the period depends on only the length of the pendulum. The period is also said to remain constant, even when the amplitude of the oscillation decrease (probably due to air resistance). This result was first obtained by Galileo, who noticed a swinging lantern and timed the oscillations by his pulse (clocks had not been invented). He found the period remained constant even though the swing gradually diminished in amplitude.

The equation I will use to determine the acceleration due to gravity at the earth’s surface is:

This means that I will need to find the period of oscillation for several different lengths of the pendulum, and then using this equation, I will be able to determine g.

Variables

The independent variable will be the length of the pendulum and the dependent variable will be the period of oscillation that is how long it takes for the pendulum to complete one oscillation.  

The period of oscillation of a pendulum depends on three factors:

  1. length of pendulum
  2. amplitude of displacement
  3. acceleration due to gravity

The gravitational field strength of earth varies slightly depending on where you are. However, the experiment is being conducted in the same place and so this will have no affect as a variable.  

The length of the pendulum will be the independent variable in this experiment. At first I planned to measure the period of oscillation for a variety of lengths and then plot a graph of T in relation to √L, the gradient of which would be 2π / √g.

This however, would not produce a linear graph and so calculating the gradient would be more difficult and ambiguous than if using a linear graph. So for the theory of the simple pendulum:

                        To obtain a linear graph, take the squared equations

which enables g to be deducted when plotting T² against L. The gradient of this graph is 4π² / g. Therefore the value of g can be calculated by 4π² / gradient.

This equation doesn’t involve mass, amplitude of displacement. It doesn’t take into account air resistance as this is negligible.

The equation doesn’t take into account mass, because it doesn’t have affect on the period of the pendulum. I will use the same pendulum throughout my experiment nonetheless.

As the amplitude of displacement is increased the gravitational potential energy of the bob also increases, because the height is increased. Therefore the kinetic energy must also increase by the same amount. This would imply that the pendulum will oscillate faster as the velocity has increased, due to increase kinetic energy. However, the increase in displacement amplitude meant that height of the pendulum bob increased and so distance the pendulum bob oscillates has also increased. Therefore the period will remain the same for this increased amplitude of displacement to a smaller displacement amplitude, as although the speed of the oscillating bob has increased, the distance it moves through also increases.

Preliminary Experiment

Before I carried out the experiment to obtain results on which to form my final conclusion I carried out a preliminary experiment. This helped me verify and check certain details and also finalise my method. I was able to identify the errors and minimise them as much as possible in order to carry out an effective and conclusive experiment.  

I am going to use a simple pendulum and count the time taken for the pendulum to complete a set number of oscillations, so that I can calculate the period. By doing this for a range of lengths I will be able to relate my results to the above equation, by finding the gradient of the line of period² plotted against length of pendulum. This is my justification for the design of my experiment.

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Fair Test

In order to ensure that a fair test was being carried out I needed to identify what aspects of the experiment needed to be fixed to make it a fair test. The following are variables which need to be kept constant.

  • Amplitude of displacement

As long as the amplitude of displacement is 15˚ or under, the period calculated on the basis of the equation I have used above, is accurate to within ½ %. (Roger Muncaster). I found that measuring the amplitude of displacement accurately was not practical and it was very easy for human ...

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