There are many situations in which some object
finds itself in an **equilibrium position**,
at which it is subject to zero net force;
but, if the object moves away from the equilibrium position,
it experiences a force pushing/pulling it back.

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One example of this situation involves a mass attached to a spring.

Q: What happens if we move the block tox = 1? Q: What happens if we move the block tox = 2?

Right. The spring will pull the block back to the left, in the negative-x direction. If we displace the block by twice as much, the spring pulls back twice as hard.

Q: What happens if we move the block tox = -1?

Right again. This time, the spring will push back, exerting a force in the positive-x direction.

If we move the block to many different positions and measure the force exerted by the spring each time, we'll find a simple relationship:

We can write this relationship as an equation; you may have seen this before. It is sometimes called Hooke's Law.

The **k** in the equation is called the **spring constant**;
it depends on the physical nature of the spring.
Stiff springs have a large value of k,
and loose springs have a small value.

Q: What are the units of the spring constant?

Okey-dokey. Let's take this relationship to a higher mathematical level, writing it as a differential equation. We begin by using Newton's First Law to replace "force" by a combination of "mass" and "acceleration".

Now, I'll re-arrange things so that the left-hand side consists solely of a second derivative. The result should look familiar to you.

Q: What is the functionx(t)that solves this differential equation?

Yes -- you can write the solution two ways:

- in terms of sine and cosine functions or
- in terms of exponential functions with complex arguments

We're going to stick with the first option in this class, so one general solution to the motion of a mass attached to a spring is

That combination of the spring constant **k**
and the mass of the block **m**
occurs over and over again, and repeatedly writing it
inside a square-root sign is awkward;
so, physicists have come up with a shortcut: they
define the **angular frequency** of an oscillating
system to be

With this substitution, the equation of motion for a mass attached to a spring becomes

Let me stop here for a moment to summarize the important point: if an object is subject to a linear restoring force,

then its equation of motion can be expressed as a second-order differential equation

Any such object will undergo **Simple Harmonic Motion** (SHM)
if displaced from its equilibrium.
In other words, it will oscillate around the equilibrium
point in a sinusoidal manner as a function of time.

Which brings us back to this equation:

Q: What are the meanings of the underlined values? What are their units?

Let's take a look at a graph of this function to illustrate the answers.

The ** amplitude A** controls the limits of the oscillation.

The **angular frequency ω** controls how quickly the
oscillation occurs.

The **phase shift φ** controls the starting time
of the oscillation.

In summary, for our example of the block attached to a spring,

variable symbol units ------------------------------------------------------- amplitudeAmeters 2 π angular frequencyωrad/sec = ------- Period phase shiftφrad -------------------------------------------------------