Studying Oscillations

Studying Oscillations

The story goes that when Galileo was attending church one Sunday, he saw a lamp at the ceiling. The lamp was swinging, possibly from the wind. Galileo started timing the swing. Watches had not yet been invented, so he probably used the pulse on his wrist. He found that as the distance of the swing decreases, the time for each swing remained the same. Surely, if distance gets smaller, it should take less time?

It turns out that as the swing distance decreases, the velocity slows down by exactly the right amount to keep the swing time constant. Accurate and systematic measurements help us to understand oscillations better. We can do it quite easily with a piece of string, a stone, and a stop watch.

Tie one end of the string to the stone, and hang the other end on a support to make a pendulum. Pull the stone to one side, let go, and time it for 10 swings. Then divide the time by 10 to get the time for one swing. Repeat this with the stone at different distances from the rest position. You will find that the swing time is nearly the same for small angles.

Suppose this pendulum is 1 m long, and the distance of the stone from the rest position is 1 cm when you let go. Then it will take about 2 s for it to swing to the other side and back again. This time is called the period. Consider the displacement x of the stone from the rest position at a certain time t. Lets take the right side to be positive. This means that if the stone is on the right, x is positive. If it is on the left, x is negative. As time passes, the displacement changes. When you first let go, it is positive. As the stone moves to the rest position, x changes to 0. As the stone swings to the left, x becomes negative. As the stone comes back to the rest position, x goes to zero again. This repeats with every swing of the stone. If we could measure x and t and plot a graph, it would look like this:

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