Cosmology for Beginners

Simple pendulum experiment
Pendulum investigation

How fast does the force of gravity accelerate objects to the ground?

If you drop an object it will always fall at the same rate, no matter how heavy it is. This means that light objects and heavy objects land at the same time, if they are dropped at the same time and they are not affected by air resistance or wind. Try it for yourself if you don't believe it! The rate at which objects are accelerated to the surface of the Earth is known as g, the
acceleration due to gravity.

In this project, you will carry out two experiments to find out how fast objects are accelerated to the ground. The first uses direct measurement, the second a simple pendulum. There are other ways also of getting a value for g which you could investigate.

Experiment 1: direct measurement

Equipment needed:

• small ball
• stop watch
• tape measure

It isn't easy to get an accurate value for g this way - you need to find a way of starting the stop watch and dropping the ball at the same moment, and to record the time as precisely as possible when the ball hits the ground.

• measure the distance through which the ball will fall
• drop the ball, finding the time it is falling for as accurately as possible

Calculating a value for g

To calculate the value of g from these measurements, you need to know a connection between distance travelled, time taken and the acceleration. Since the ball is starting from rest, the formula you need is:

where s is the distance travelled, t is the time taken, and a is the acceleration. If distance is measure in metres and time in seconds, then acceleration will be in metres per second squared, or m/s².

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The simple pendulum

Now compare the value you obtained from a direct measurement with the value you get from a simple pendulum. A simple pendulum is just a piece of string with a mass suspended at one end which is allowed to swing freely. The Earth's gravity controls how the pendulum swings.

Equipment needed:

• string (at least 1 metre long)
• mass tied to one end of the string
• something to suspend the pendulum from
• stop watch

In the diagram on the right, the dotted line shows the rest position of the pendulum, while the solid line shows the pendulum ready to start swinging. The length of the string is L, and the object (bob) on the end of the string has mass m. You will need to count a number of complete swings of the pendulum (a complete swing is from one position of the pendulum back to that
position) and time them.

• measure the length of the pendulum
• record the mass of the bob
• let the pendulum swing a given number of times, say 10 complete swings starting with the pendulum displaced from its rest position by a small angle only
• find the time taken
• repeat this several times
• what happens if you vary the angle the pendulum starts from?

This website http://www.edumedia.fr/a255_l2-simple-pendulum-1.html has a simulation which you might find useful. You only get a short while in the demo version, but it would be possible to get some results to help with this experiment.

Calculating a value for g

The time taken for one complete swing is known as the period of the pendulum. Find a value for this. To calculate a value for g, you will need to use this formula:

Here, T is the period of your pendulum, L is its length and g is the value you are trying to find. You will need to start by rearranging the formula. How well does this value for g compare with that you obtained in the first experiment?

Pendulum investigation

Try experimenting with different lengths of string and different masses on the end of it. Does it make a difference to the period when you do this? Does it make a difference to the value you get for g?

You should find that provided you let the pendulum swing in a controlled way (meaning the starting angle is not too great), that the angle and the size of the mass do not make any difference, but the length of the string does.

Try collecting information about the period of the pendulum for different lengths of string. Tabulate your data, and plot a graph of them. What kind of graph do you get if you plot T against L? Is it a straight line? You should find you get a curve. Try
plotting T² against L instead. This should give you a more or less a straight line (as on the left). You can put your data into a spreadsheet and use this to draw a graph if you wish. If you do that, you can ask the spreadsheet to give you the equation of your line.

Alternatively, you can find the equation from the slope or gradient of the line. On the sketch graph on the left, you can see a green triangle has been added. If you divide the height of a triangle like this by the base (it doesn't matter how big the triangle is provided the height and base are vertical and horizontal respectively), that will give you the gradient of your line. (Note
that your line should go through the origin - the point (0,0)).

If you rearrange the equation given above for a simple pendulum so that T² is on the left-hand side, then the value which multiplies L corresponds to the gradient of your graph. Once you know this, you can work out a value for g from it by using the equation given above. You will need to rearrange this equation so that you have T² on
the left hand side first, and then the gradient is the same as the values which multiply the L. Does this give you more or less the same value for g as you obtained earlier?

The usual value given for g is about 9.8. How close is your value? How do you think you might improve the accuracy of your value?