I asked the class what is meant by “simple”. The responses were strange and wrong! Here is how we should explain it, in my humble opinion. To begin, we must try to avoid saying we cannot explain a topic as they lack the necessary mathematics background. Instead, we can describe the math so that the results appear reasonable. Just because we cannot give rigorous proofs does not mean that we can skip the mathematical ideas essential to the understanding of what we are saying.

 

For pendulum motion, the force on the pendulum is F = mg sinθ, where θ is the angle from the vertical. They must understand this equation. Then we say that for small θ,

 

sin θ ≈ θ.

 

Although we cannot prove this, we can show that this is reasonable by drawing the sin curve and noting that it starts out like a straight line. For small angles, the equation is indeed simple, as the equation is

 

F = mgθ.

 

For large angles, of course, it is not simple. Of course, the angles are measured in radians.

 

Let us look at a pendulum.

A mass is attached to the ceiling by a light string of length L. If the displacement from the vertical is θ, the net force F on the string is perpendicular to the string. This is because the component of gravitation parallel to the string is equal to the tension of the string. We see this if the mass is larger than the maximum tension, the string will break.

 

The mass will swing back and forth, if we neglect things like air resistance and heat in the string. This is harmonic motion.

 

Let us look at the equations, and then we will explain what is “simple”. The force is

 

 

Let x be the position of the mass with respect to the equilibrium position. Look at the triangle: one side L, the other side x, and the third side, the hypotenuse, the vertical dashed line. If the angle is small, then L is almost equal to the hypotenuse. Recall that sin is defined as opposite divided by hypotenuse. The opposite is x. We get

       

sin θ ≈ x / L.

 

The force equation becomes

 

       F = -mgx / L

 

This equation can easily be solved. This is why it is called simple. If the angle is large, the motion is not simple.

 

Check this yourselves with your calculators. Use radians. You see that sin(.1) = 0.0998. Check other values of sin. We see that for small angles, indeed

 

sin θ ˜ θ.

 

By the way, 0.1 radians is about 5.7°.

 

Newton’s Law of motion is F = ma. The equation for the pendulum is then

 

       ma = -mgx / L

 

The masses cancel, and the equation is

 

       a = -gx / L

 

The acceleration of the pendulum depends on –x. When x = 0, the mass is at the equilibrium point. There is no force on the mass, and so no acceleration. When x is large, the acceleration is also large, in the opposite direction. This is the meaning of the minus sign. The units of acceleration are m/s². The right side has the same units, as the units (meters) of x cancel the units of L.

 

We note with surprise that the acceleration does not depend upon mass. Take two pendulums, one with a light wooden mass and the other with a heavy steel mass. Does it make sense that both have the same accelerations? They do!

 

The reason is that the force of gravity depends upon the mass. The acceleration is force divided by mass, and so the masses cancel. Einstein noted this strange thing. The mass due to gravity is the same as the inertial mass in Newton’s equation. This coincidence was the basis of his theory of gravitation, called the General Theory of Relativity.

 

Let us analyze this some more. It is very easy to do this analysis with calculus. The challenge is to explain it to students without calculus.

 

Draw a graph of the position x with respect to time t. It gets bigger and smaller, bigger and smaller. A nice function that will do this is the sin function. It starts with zero, goes to 1, down to -1, back to zero again and again.

 

Here is a function x =A sin(ωt). A is the amplitude, units meters, as is x. The argument ωt of the sin function is a number (no units), as the units of ω are per second, or /t.

 

When ωt is 90°, or π/4 radians, sin is one.

 

A full cycle is when ωt is 2π, which is when t is the period T.

 

We see that

 

T = 2π / ω

 

Let us agree that the motion of the pendulum is given by x =A sin ωt, as it is a simple function that goes up and down. This looks reasonable, but we are not going to give a formal proof here.

 

What would the velocity v be? Remember that v = Δx/Δt. The change of a function is the slope. Look at the curve. It starts out with a slope of 1, then, after a quarter period, the sin is a maximum, the slope is 0. It then decreases. The slope is the same function as the sin, just phase shifted by 90°, which is the cos function, which is the sin function, starting at 90° instead of 0. Here is the same curve, shifted to the right by a quarter cycle:

 

Now let us find the acceleration, which is Δv/Δt. The slope is again phase shifted by 90°, which is negative our original sin function.  Here is the curve shifted again by a quarter cycle, or a half cycle of the original curve:

 

We see that this curve is the same as the curve for x, except for a minus sign:

 

a = -Aω²sin ωt

 

The units must balance on both sides of the equation. Acceleration a is m/s². The amplitude A is in meters. We have to multiply by ω² in order to get /s², per second squared.

 

The above equation is

 

        a = -ω²x

 

Compare this with a = -(g/L)x, we see that

       

       ω² = g/L,

 

or

This is the equation for the period of a pendulum. If L is large, so is the period. Imagine a large clock. It sl-ow-ly ticks. If g is large, it will move rapidly.

 

We see something else important. We recall that the equation is

 

       F = -mgx/L

 

This is the equation for a spring, Hooke’s Law:

 

       F = -kx

 

It has the same form as the simple pendulum, with k = mg/L. This means that the period is

This makes sense, for if k is large (a strong, stiff spring), then the period is small – it vibrates rapidly. If the mass is large, then the period is small.

 

The periodic harmonic motion of a pendulum and a spring are the same.

 

To summarize, the tools we used to clarify this are just basic understanding of periodic functions and dimensional analysis. All equations must have the same units on both sides of the equation. If the left is in meters, the right side must be meters.

 

Whenever you do problems, be sure that you check that the units are correct, not only that the numbers are correct.