# Oscillations

### swing motion equilibrium child

An oscillation is a particular kind of motion in which an object repeats the same movement over and over. It is easy to see that a child on a swing and the pendulum on a grandfather clock both oscillate when they move back and forth along an arc. A small weight hanging from a rubber band or a spring can also oscillate if pulled slightly to start its motion, but this repeated motion is now linear (along a straight line). On a larger scale, you can notice oscillations when bungee jumpers fall to the end of their cords, are pulled back up, fall again, etc. Actually, oscillations are all around us, even in the pages of this book.

Anything, no matter how large or small, can oscillate if there is some point where the object is in stable equilibrium. Stable equilibrium means that an object always wants to return to that position. Suppose you placed a marble at the exact center inside a very smooth bowl. If you tap the marble slightly to move it a small distance, it rolls back towards the center, overshoots, rolls back, overshoots, etc. The marble is oscillating as it continues to return to center of the bowl, its point of stable equilibrium. If you think of the marble and the bowl as a "unit," you can see that the "unit" stays together even though the marble is oscillating (unless you tap the marble so hard that it flies out of the bowl). This is the reason for using the term stable.

On the other hand, what if you turned the bowl over and tried the same experiment by placing the marble on top at the center. You might succeed in balancing the marble for a short time, but eventually you will touch the table or a breeze will move the marble a small amount and it will fall. When this happens, the "unit" of marble and bowl comes apart and no oscillation can happen. In this case, the center of the bowl would be a point of unstable equilibrium, since you can balance the marble there, but the marble cannot return to that point when disturbed to keep the "unit" from disintegrating.

For the motion of a child on a swing, the bottom of the arc (when the swing hangs straight down) is the point of stable equilibrium. The point of stable equilibrium for a weight on the rubber band is the location at which the weight would hang if it was very slowly lowered. In either case, an oscillation occurs when the object (child or weight) is moved away from stable equilibrium. If we pull the swing back some distance the child will move toward the bottom of the arc. At the instant the swing is at the point of stable equilibrium, the child is moving the fastest since as the swing proceeds up the arc on the other side, it slows down. The higher the swing was when the motion was started, the faster the child moves at the bottom. The swing overshoots stable equilibrium and the child rises to the same distance on this side of the bottom as on the starting side. For a brief instant the swing will stop before the swing begins to retrace its path, traveling in the other direction.

This simple example demonstrates several properties shared by all oscillations: 1) The point of stable equilibrium is the center of the oscillating motion since the object moves the same distance on either side. That distance is called the amplitude of the oscillation. 2) At either end of the motion, the object stops briefly (slowest location) while the fastest location is when the object is just passing through the point of stable equilibrium. 3) The energy that an object has when it is oscillating is related to the amplitude. The larger the amplitude, the larger the energy.

Oscillations also have two very specific properties regarding time. Every oscillation takes a certain amount of time before the motion begins to repeat itself. Since the motion repeats, we really only need to worry about what happens in one cycle, or repetition of the oscillation. If we pick any point in the motion and follow the object until it has returned to that same point ready to repeat, then the oscillation has completed one cycle. The amount of time that it took to complete one cycle is called the period, and every cycle will take the same amount of time. Suppose for the child on a swing we pick the point at the bottom of the arc. When the swing moves through that position, we start a timer. The child will swing up, stop, swing back down through the bottom (but traveling in the other direction), swing up, stop, and swing back through our point. Now the child has returned to our starting point and the motion is about to repeat, so we stop our timer. The curious thing is that even when the amplitude is changed, the period stays the same. This is because even though the child moves faster when pulled higher to start the motion, the swing also has farther to travel to complete a cycle.

The other time property is called the frequency, which tells how often the motion repeats. This really gives the same information as the period since if it takes 0.5 second for 1 cycle, then the frequency will be (1 cycle)/(0.5 second) = 2 cycles per second. Often a unit called the Hertz (Hz) is used to represent a cycle per second. The cycles per second should sound familiar because the magnitude, or amplitude, of the electrical current in most households oscillates at 60 cycles per second. The time properties of an oscillation are very important since they control how best to add energy to the motion.

The child on a swing, weight on a spring, and bungee jumper on an elastic cord are all types of oscillations which we can see with our eyes. However, if you kick a rock (disturb it) and it does not disintegrate, then the atoms within the rock must be in stable equilibrium. The atoms within the rock are therefore capable of oscillating. Small oscillations are actually occurring all the time in every seemingly solid substance, including the rock and this page. We cannot see this motion, but we do feel it. The larger the amplitude of those small oscillations of the atoms, the hotter the object, and that is something we can detect directly.

James J. Carroll

## KEY TERMS

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Cycle

—One repetition of an oscillation as an object travels from any point (in a certain direction) back to the same point and begins to move again in the original direction.

Equilibrium

—The condition when all the influences (forces) trying to move an object are balanced, at least for an instant.

Frequency

—The number of cycles of an oscillating motion which occur per second. For example, if the period for one cycle is 0.5 second, then the frequency is (1 cycle)/(0.5 second) = 2 cycles per second = 2 Hertz.

Period

—The amount of time it takes for one cycle of an oscillating motion.

## User Comments

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almost 2 years ago

#confused. I was trying to use this to cheat on a test but I failed lol :(((