Question

How to form an equation for simple harmonic motion?

Answer 1

All simple harmonic motion is sinusoidal. This can best be illustrated visually. As you can see from our animation (please see the video at 01:34), a mass on a spring undergoing simple harmonic motion slows down at the very top and bottom, before gradually increasing speed again as it approaches the center. It spends more time at the top and bottom than it does in the middle. Mathematically, any motion that has a restoring force proportional to the displacement from the equilibrium position will vary in this way.

Because of that, the main equation shown below is shaped like a sine curve. It says that the displacement is equal to the amplitude of the variation, A, otherwise known as the maximum displacement, multiplied by sine omega-t, where omega is the angular frequency of the variation, and t is the time. This displacement can be in the x-direction or the y-direction, depending on the situation. A vertical mass on a spring varies in the y-direction sinusoidally. A horizontal mass on a spring varies in the x-direction sinusoidally. A pendulum has such a variation in both directions
horizontal on a smooth tabletop.

If you imagine pulling a mass on a spring and then letting go, it will bounce back and forth around an equilibrium position in the middle. Like with all simple harmonic motion, the velocity will be greatest in the middle, whereas the restoring force (and therefore acceleration) will be greatest at the outside edges (at the maximum displacement). Another example of simple harmonic motion is a pendulum, though only if it swings at small angles.

Equations
There are many equations to describe simple harmonic motion. The first we're going to look at, below, tells us that the time period of an oscillating spring, T, measured in seconds, is equal to 2pi times the square-root of m over k, where m is the mass of the object connected to the spring measured in kilograms, and k is the spring constant (a measure of elasticity) of the spring. The time period is the time it takes for an object to complete one full cycle of its periodic motion, such as the time it takes a pendulum to make one full back-and-forth swing.

Equation for Simple Harmonic Motion
equation for harmonic motion
All simple harmonic motion is sinusoidal. This can best be illustrated visually. As you can see from our animation (please see the video at 01:34), a mass on a spring undergoing simple harmonic motion slows down at the very top and bottom, before gradually increasing speed again as it approaches the center. It spends more time at the top and bottom than it does in the middle. Mathematically, any motion that has a restoring force proportional to the displacement from the equilibrium position will vary in this way.

Because of that, the main equation shown below is shaped like a sine curve. It says that the displacement is equal to the amplitude of the variation, A, otherwise known as the maximum displacement, multiplied by sine omega-t, where omega is the angular frequency of the variation, and t is the time. This displacement can be in the x-direction or the y-direction, depending on the situation. A vertical mass on a spring varies in the y-direction sinusoidally. A horizontal mass on a spring varies in the x-direction sinusoidally. A pendulum has such a variation in both directions.

Main Equation for Harmonic Motion
main equation with sine
This equation has a sine in it, and a sine graph starts at zero. Using this equation is like starting your mathematical stopwatch in the middle of a pendulum swing: t = 0 is in the center of the oscillation. If, on the other hand, you replace sine with a cosine, then the equation is still correct; you're just starting to measure time at the maximum displacement instead.
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