How Simple Harmonic Motion Really Works—and the Mathematics Behind It

Introduction

In high-school mathematics and physics, trigonometric functions are extremely important tools. The two that matter most—and that we use most often—are the sine function and the cosine function .

Are you asking whether I am deliberately excluding tan x? Yep. I am. (On purpose.)

These two trigonometric functions have been enormously important to the development of mathematics and physics, especially when it comes to describing waves.

In the previous few articles, I discussed the basic ideas of differentiation and integration, along with their mathematical and physical meaning, in “A Gentle Introduction to Differentiation” and “A Gentle Introduction to Integration.”

And in “Differentiating Trigonometric Functions,” I went one step further and explained where the differentiation formulas for sine and cosine come from.

In this article, we are going to see what roles these two trigonometric functions play in simple harmonic motion in high-school physics.

So, you are saying everything before this was basically setting up this article?

Exactly!! That is exactly what I was doing!

There are many kinds of simple harmonic motion. Teachers may even discuss examples such as circular-orbit problems involving celestial bodies or floating objects bobbing in water. Here, though, I want to start with two relatively simple systems—a horizontal spring system and a vertical spring system—and use them to see what sine and cosine are really doing inside high-school simple harmonic motion.

Horizontal Spring System

Basic Derivation

Consider a spring on a smooth horizontal surface. Let its natural length be l and its spring constant be k. One end is fixed to a wall, and the other end is attached to a block of mass m, as shown below:

This is a fairly common setup in high-school physics. You will see it in chapters on simple harmonic motion and in energy problems involving elastic potential energy.

Intuitively, the block is pulled back and forth by the spring and oscillates along the surface. Now let us try to describe that physical behavior with mathematics.

Let time be t and let the spring extension be x(t). Assume that every deformation stays within the elastic limit, so the spring does not suffer elastic fatigue. Here x(t) is also the block’s displacement relative to the spring’s natural length, so its sign matters.

From Newton’s second law, the force acting on the block can be written as follows:

Here, x(t) represents the block’s displacement function.

From the basic calculus ideas used in kinematics, differentiating displacement once with respect to time gives x'(t), the block’s velocity function. Differentiating velocity once more gives x''(t), the block’s acceleration function.

Hooke’s law, on the other hand, tells us that the spring force is proportional to the spring’s extension. The force on the block can therefore be written as follows. For convenience, we include the direction directly in the sign:

Combining the two equations above gives the equation of motion satisfied by the block:

Looking at this equation, we can observe that it is, in fact, an equation (yes, obviously), but it contains both the variable itself and the second derivative of that variable.

An equation that describes the relationship between a variable and its derivatives is called a differential equation (D.E.).

Because this differential equation involves only one independent variable, it belongs to the class of ordinary differential equations (O.D.E.s).

A general solution to this differential equation can be written as follows:

Here, c₁ and c₂ are constants. We still need the block’s initial conditions before we can determine their values.

If you want to know how this ordinary differential equation is actually solved, you can look into courses such as Engineering Mathematics or Differential Equations, which are commonly offered in university science and engineering programs.

To determine these two constants, let us look at a concrete example.

Suppose the block initially sits where the spring is neither stretched nor compressed, so the spring remains at its natural length. In other words, .

Now suppose the block also has an initial velocity v, so .

The equation describing the block can then be written as follows:

We can then solve for the block’s motion as follows:

If you are interested, try substituting the general solution x(t) back into the differential equation yourself. You will need the chain rule, together with the trigonometric differentiation formulas discussed in the previous article!

Other Properties of Motion

Once we have the position function for the simple harmonic motion, differentiating it once gives the block’s velocity function:

Using what we know about the range of the sine function together with the velocity function for the simple harmonic motion, we can see that the block can move no farther than and . We call these two positions the “turning points” of the horizontal spring system.

We can also see that the block’s velocity is zero at the turning points.

On the other hand, when the block passes through the origin, its speed reaches the maximum value .

At this point you might be thinking, “Wait a second. You used a simplified special case to explain all of this. How do you know the same conclusions are always true?”

Fair point!

All right, let us return to the general solution for the block’s motion. I am putting it here again so you do not have to scroll all the way back up XD.

Using a trigonometric combination identity, we can rewrite the expression as follows:

Here α represents a particular phase determined by the object’s initial position. Since I only want to examine the overall motion here, we can leave the exact value of α aside for now.

We can see that the block oscillates back and forth around the origin x=0. We call x=0 the “equilibrium position.”

If the block starts at rest exactly at the equilibrium position, it will remain there forever.

But if the block starts away from equilibrium—or starts at equilibrium with some velocity—the spring force makes it oscillate back and forth.

That is also where the name “equilibrium position” comes from. If the block is placed there at rest, it really can stay there indefinitely. Otherwise, it oscillates around that point, which acts as the center of its equilibrium state.

Next, differentiating the expression above once with respect to time gives the block’s velocity function:

We find that when x=0, reaches its maximum value; when , also reaches its maximum value.

In other words, the block moves fastest as it passes through equilibrium, while its speed becomes zero at the two turning points.

Isn’t that exactly the same conclusion we reached earlier from the special case? Sometimes, if you want to understand a problem quickly, choosing a convenient special case to simplify it is incredibly useful!

Relationship Between Simple Harmonic Motion and Uniform Circular Motion

From the rewritten equation of motion, we can also see that the block’s motion can be viewed as the projection of uniform circular motion onto either axis.

The reason is that both the position function and the velocity function of the block can be expressed as a single sine or cosine function.

That means we can treat the center of the uniform circular motion as the coordinate origin. At any moment, the corresponding sine and cosine values let us project the position and velocity onto the x- and y-axes and use those projections to describe the spring system.

This is why, when students do not yet have a background in ordinary differential equations, many high-school physics teachers introduce the statement: “Simple harmonic motion can be viewed as the projection of uniform circular motion onto an axis.” The idea comes from looking at the position and velocity functions that emerge from the differential-equation solution.

Sounds abstract, right? Let us use a picture to make it a little clearer.

The figure above shows uniform circular motion. We can write the angle as ωt+φ, where ω is the angular speed of the circular motion.

For a horizontal spring system, the picture is like projecting the uniform circular motion onto the x-axis. The position of the simple harmonic motion can therefore be viewed as the radius R multiplied by the cosine of the angle ωt+φ.

Velocity and acceleration can also be handled through the same analogy. But because most high-school physics courses already cover those details, I will not spend too much time repeating what you can find in the textbook : D.

One thing is worth being precise about: we are using the projection of uniform circular motion onto an axis as a way to understand simple harmonic motion. The two have an analogical and mathematical relationship; this does not mean the two physical motions are literally the same thing.

It is only a small difference in wording, but physics is the kind of subject where that small difference matters!

Period Formula

Based on the idea that simple harmonic motion can be represented by the projection of uniform circular motion onto an axis, one complete oscillation corresponds to one complete revolution in the circular-motion picture.

Comparing the coefficients in the position functions for uniform circular motion and simple harmonic motion gives the following relationship:

If one revolution of the uniform circular motion corresponds to one complete simple harmonic oscillation, then we can use the period of the circular motion to find the period of the simple harmonic motion. For uniform circular motion, the period is:

This immediately gives the period of the block’s simple harmonic motion:

This is the simple-harmonic-motion period formula that high-school physics teachers love telling students they absolutely have to memorize!

Doesn’t it give you a little “Whoa—so that is where it comes from!” feeling : D?

Vertical Spring System

Basic Derivation

Now let us hang the spring from the ceiling and attach the same kind of block to the other end. This time, in addition to the spring force, the block also experiences its own weight, as shown below:

Assume there is no air resistance. Let the spring extension still be x(t), and let the gravitational acceleration be g. The block then satisfies the following equation of motion:

The general solution can be written as follows:

Other Motion Analysis

Using the same trigonometric combination technique, we can rewrite the expression as follows:

We find that the block still oscillates back and forth, but the center of the motion shifts from the origin x=0 to . We call the “equilibrium position.” This value is exactly the spring extension required by Hooke’s law to balance the block’s weight.

Using the same reasoning as before, we obtain the same period formula:

This means the block’s period does not depend on the value of gravitational acceleration, and the formula is identical to the one for the horizontal spring system.

Notes

More generally, if forces other than the spring force and gravity act on the block, its equation of motion can be written in the following form:

Here F(t) represents a general external force, whose exact form depends on the situation or the problem.

In particular, if air resistance is included, the motion is no longer ideal simple harmonic motion. The equation of motion and its solution become more complicated, and to keep this article from wandering too far away from its main point, I will stop here : D.

Conclusion

Whether we consider a block attached to a spring on a smooth horizontal surface or a block hanging from a ceiling spring with air resistance neglected, the block oscillates regularly in simple harmonic motion. Its speed is greatest at the equilibrium position and zero at the two turning points.

For the horizontal spring system, the equilibrium position is where the spring is not stretched or compressed—that is, where the spring remains at its natural length l.

For the vertical spring system, the equilibrium position is where the spring has stretched enough under the block’s weight to satisfy Hooke’s law—in other words, after the spring has extended by .

One thing these two ideal systems have in common is that the oscillation period does not depend on gravitational acceleration.

You may have noticed that these are exactly the same conclusions taught in high-school physics. We took a more complicated road, but arrived at the same place—and that is one of the things that makes physics such an interesting subject!

Next time you encounter a simple harmonic motion problem, try connecting the ideas here with what you learned in class. I hope it makes you like mathematics and physics just a little more : D.