Lesson Video: Oscillatory Motion Science

Video Transcript

In this video, we will learn how to describe the motion of oscillating objects. An example of an oscillating object is a playground swing. At first, the swing will hang straight down, with the rope of the swing vertical. If we left the swing like this, it wouldn’t move. It would stay in this position forever. We call this position the equilibrium position of the swing.

But what would happen if we were to give the swing a push? If we were to push on the back of the swing, the swing would start to move to the right of the equilibrium position. Eventually, it will reach its rightmost point. Then, it’ll change direction and start moving back towards equilibrium. Once it reaches equilibrium, it’ll keep moving towards the left. Eventually, it’ll reach the leftmost point.

At this point, it’ll change direction again and move towards equilibrium. Once the swing reaches equilibrium, it keeps moving back towards the right, and the whole process starts again. The swing will continue to move back and forth repeatedly. Each time it swings, it passes through its original equilibrium position, where the rope is vertical. When an object repeatedly moves back and forth through an equilibrium position, we can describe its motion as oscillatory.

Lots of different objects can have oscillatory motion. In physics, we often talk about an object called a pendulum. A pendulum consists of a bob suspended from a piece of string. A pendulum is very similar to the playground swing that we just talked about. When left to its own devices, the pendulum will hang straight down in its equilibrium position. But if we were to move the bob to one side of the equilibrium position and then let go, the pendulum will begin to swing back and forth through its equilibrium position. In other words, the pendulum will oscillate, just like a playground swing.

Pendulums can oscillate back and forth for a long time, so they’re useful tools for observing oscillatory motion. In particular, we can look at the displacement of the pendulum’s bob at different times in its oscillation. The displacement of the pendulum’s bob refers to its distance from the equilibrium position in a particular direction, left or right. Let’s imagine that the pendulum is stationary and hanging straight downwards. Here, the pendulum is at its equilibrium position. So, its displacement from its equilibrium position is zero.

Now let’s imagine that we were to move the bob to the left-hand side like this. We have now given the bob some displacement to the left of the equilibrium position. If we were to let go of the bob and allow the pendulum to oscillate, the pendulum would never move further to the left than this. It’ll return to this point, but it’ll never go past it. So, this is the pendulum’s maximum displacement to the left. When the pendulum swings, it’ll move through the equilibrium position to the right until it reaches its rightmost point. Here, it has its maximum displacement to the right.

There is something important we should notice about the positions we have labeled maximum displacement. The displacement of the rightmost point has the same magnitude as the displacement of the leftmost point. So, the maximum displacement of the pendulum has the same magnitude on either side of the equilibrium position. For the pendulum to do one complete oscillation, it has to travel back and forth through every point of its swing and return to the position that it started from.

For example, if we wanted to observe one complete oscillation of the pendulum starting from its maximum displacement to the left, we would watch the pendulum move through the equilibrium position to the right until it reaches its rightmost point. Then, we’d watch it change direction, move back through equilibrium position to the left until it reaches the point where it started from. At this point, the pendulum has completed one single oscillation.

So far, we have seen that when an object oscillates, its displacement changes in a regular and repeated way. The speed of the pendulum also follows a repeated and regular pattern. The pendulum has the greatest speed when it passes through the equilibrium position. It doesn’t matter whether the pendulum is moving to the left or to the right. When it’s at the equilibrium position and so has zero displacement, it always has the same maximum speed.

When the pendulum has the maximum displacement from the equilibrium position, its speed is always zero. The pendulum has to have zero speed for a very brief moment in order for it to change direction and move back towards equilibrium. Again, it doesn’t matter whether the pendulum is to the left of the equilibrium or to the right. When its displacement has the maximum possible magnitude, its speed is always zero.

Since the speed and displacement of the pendulum follow a regular and repeated pattern, it makes sense that the time it takes for the pendulum to swing does not change. If the pendulum isn’t affected by air resistance, it will take the same amount of time for the pendulum to complete each oscillation. Again, the direction of the pendulum’s motion does not affect the amount of time it takes for the pendulum to swing. For example, let’s say it takes one second for the pendulum to complete half an oscillation and move from its leftmost point to its rightmost point. It would also take the pendulum one second to complete half an oscillation traveling in the opposite direction from its rightmost point to its leftmost point.

So, by looking at the oscillation of a pendulum, we have seen that oscillation is a regular, repeated motion through an equilibrium position. We have also seen that the displacement of the pendulum, the speed of the pendulum, and the time taken for the pendulum to complete an oscillation all follow regular and repeated patterns.

We need to be careful not to confuse oscillatory motion with periodic motion. Periodic motion is also regular and repeated. But periodic motion does not have all of the same properties as oscillatory motion.

For example, consider a bouncy ball. If we placed the ball on the ground, it wouldn’t move. So, the ground is the ball’s equilibrium position. Let’s imagine that we drop the ball from a height. It’ll fall to the ground. Then, it’ll change direction and start to move upwards. If the ball doesn’t lose any energy, it’ll return to its initial maximum height and continue to bounce in this way. This is periodic motion, but it’s not oscillatory. The ball has an equilibrium position, but it’s impossible for the ball to pass through it. The ball is only ever above the ground and never below it. This is not consistent with our earlier descriptions of oscillation.

Oscillation is repeated motion through an equilibrium position. Plus, in order for the ball to reverse direction when it hits the ground, it must momentarily have zero speed. So, when the ball is at its equilibrium position, its speed is zero. But an oscillating object has its maximum speed when it passes through the equilibrium position. So, for our bouncy ball, the relationship between speed and displacement is not consistent with oscillatory motion.

However, there are objects other than pendulums that can have oscillatory motion. An example of such an object is a spring. Here we have a spring, where one end is fixed in place and the other end is free to move. At the moment, this spring is at rest, so the spring isn’t moving, and nothing is being done to the spring to change its length. This is the spring’s equilibrium position. If the spring is left like this, it’ll stay like this. But we can apply forces to the spring to change its length.

If we push on the free end of the spring, the spring will get shorter. We say that the spring is compressed. When we compress the spring, the spring will start to exert a reaction force to resist being compressed. When we let go of the spring, this force will cause the spring to spring back to its original length. But the spring won’t just spring back to its original length. It’ll actually keep moving, stretching itself out, and making itself longer. Again, once the spring has begun to stretch, it’ll start to exert a force on itself to resist this stretching. This will cause the spring to move back to its original length. But again, it won’t stop once it’s reached its original length. It’ll keep moving until it has returned to its compressed position, and the process will continue.

This is an example of oscillatory motion. Here, we have drawn one complete oscillation of the spring. We know that the spring is oscillating because the spring follows a regular repeated pattern of motion through an equilibrium position. The speed and displacement of the spring also have all the properties of oscillatory motion. When the spring is fully compressed, it has maximum displacement to the left of the equilibrium position. When the spring is fully stretched, it has a displacement of the same magnitude to the right. And the speed of the spring is greatest when it is moving through equilibrium. The spring momentarily has zero speed when it has its maximum displacement.

The two examples of oscillatory motion that we have discussed in this video, the spring and the pendulum, may seem very different, but we can compare the two. Compressing the spring is equivalent to moving the pendulum bob to the left of its equilibrium. When the pendulum is released, it moves through equilibrium and then keeps moving towards the right. This is just like a spring, which returns to its original length but then keeps moving until the spring is stretched out. We know that the pendulum will then change direction and pass through equilibrium before moving back towards the left. This is just like the spring returning to its original length before compressing again.

Now we know how to identify oscillatory motion, let’s take a look at some example questions.

A pendulum swings freely from left to right. At which two of the positions shown is the pendulum equally far from its equilibrium position? (A) I and III, (B) I and IV, (C) II and III, or (D) II and IV.

To answer this question, we need to determine which two positions of the pendulum are equally far from the equilibrium position. First, let’s make sure we know what is meant by the pendulum’s equilibrium position. The equilibrium position of an object is the position where the object will not move unless a force acts on it to make it move. A pendulum is at its equilibrium position when the bob is hanging straight down, with its string vertical. This corresponds to position II in the diagram.

So, we need to find out which two positions are equally displaced from position II in the diagram. We can rule out any answer options which contain position II straightaway, since position II cannot be equally far away from itself and another point. So, we know that answer options (C) and (D) aren’t correct.

To help us decide which of the remaining options is right, we can label the displacement of positions I, III, and IV on the diagram. Recall that the displacement of the pendulum from its equilibrium position is its distance from equilibrium in a given direction. Let’s label the displacement of position I. Position I has a displacement to the left of the equilibrium. The magnitude of this displacement is shown by the length of this blue arrow. Both positions III and IV have a displacement to the right of the equilibrium, which again we can represent like this.

To compare the magnitude of the displacement at each position, we simply compare the length of these arrows. We can see that the arrows corresponding to positions I and IV are the same length. But the arrow corresponding to position III is much shorter. This tells us that positions I and IV have the same magnitude of displacement from the equilibrium position. So, the correct answer to this question is option (B).

Let’s take a look at another example.

A pendulum swings from left to right and back again. The pendulum oscillates without dissipating energy. At which of the following positions is the speed of the pendulum greatest? (A), (B), or (C).

In this question, we need to work out which of these three positions shows the pendulum when it has the greatest speed. We are told that the pendulum oscillates without dissipating energy. This means that the speed of the pendulum will continue to change according to the same repeated pattern, regardless of how many times the pendulum has oscillated.

Option (A) shows the pendulum at its equilibrium position. Option (B) shows the pendulum when it has its maximum displacement to the left. Option (C) shows the pendulum when it has its maximum displacement to the right.

We can recall that the speed of an oscillating object is related to the distance of the object from the equilibrium position. When an object has its maximum displacement, either to the left or to the right of equilibrium, its speed is momentarily zero. This is because the object is about to change direction and start moving back towards equilibrium. On the other hand, any time an object is passing through the equilibrium position, its speed always has the maximum possible value. So, we can say right away that the correct answer to this question is option (A), as option (A) shows the pendulum passing through its equilibrium.

In fact, it doesn’t really matter what positions are shown in options (B) and (C). Any two nonequilibrium positions would still always correspond to a lower speed. So, the correct answer to this question is option (A).

Let’s look at one more example, this time about a spring.

A spring attached to a wall is pushed until it reaches the red line, and then it is released. The diagrams show how the length of the spring changes after it is released. The spring oscillates without dissipating energy. The blue line shows the equilibrium position of the spring. How does the distance between the red and blue lines compare to the distance between the green and blue lines?

The diagram is showing us the oscillatory motion of the spring. The spring is first compressed to the red line. Then, it’s allowed to oscillate back and forth through the equilibrium position, the blue line. The furthest distance that it stretches to is indicated by the green line. The red and green lines represent the furthest nonequilibrium positions of the spring. We are told that the spring does not dissipate energy while it oscillates. So the spring will continue to oscillate between these two positions.

To answer this question, we need to recall that when an object undergoes oscillatory motion, the magnitude of its maximum displacement should be the same on either side of its equilibrium position. When the spring is at its equilibrium position, the end of the spring is in line with the blue line. When the spring is maximally compressed, it has its maximum displacement to the left of equilibrium. This happens when the end of the spring is in line with the red line. When the spring is maximally stretched, it has its maximum displacement to the right of equilibrium. This happens when the end of the spring is in line with the green line.

We know that the magnitude of the spring’s maximum displacement is the same on either side of its equilibrium position. So, we’re ready to answer the question of how the distance between the red and blue lines compares to the distance between the green and blue lines. We know that the distance between the blue line and the red line is the same as the distance between the blue line and the green line. The answer to this question is therefore “The distances are the same.”

Now let’s finish up this video by summarizing some key points. Oscillatory motion is regular, repeated motion through an equilibrium position. The magnitude of the maximum displacement of an oscillating object is the same on either side of the equilibrium position. The speed of an oscillating object is zero at the furthest nonequilibrium positions and maximum at the equilibrium position. The time it takes for an oscillating object to move through one complete oscillation is the same every time. Examples of objects which can undergo oscillatory motion are pendulums and springs.