Newton’s Second Law of Motion

Now, imagine that we have another mass hanging from a spring, which is attached to the ceiling. This mass is then pulled down a distance , below the equilibrium point, and released. By Hooke's Law, the spring will be pulled back up, and after reaching it's highest point, start to travel back down.

If the mass continues this motion, without any outside influence, it's known as Free Motion. During this motion, there is acceleration acting on the mass to keep it in motion. This acceleration can be readily found in Newton's Second Law of Motion using

. Where is the force of the spring acting on the mass and is the acceleration of that mass due to the force acting on it.

From Calculus, we know that or . If we substitute this into our equation

from the section on Hooke's Law, we find . However, since this motion is always in the direction of the spring's force, our equation becomes

. Expanding this, and solving for we get

. Knowing that , we can simplify our equation to end with .

The Differential Equation of Free Motion

Finally, if we set the equation above equal to zero, we end up with the following:

Since our leading coeffiecient should be equal to 1, we divide by the mass to get:

If we set , we'll have our final form of this equation:

The above equation is known to describe Simple Harmonic Motion or Free Motion.

Initial Conditions

With the free motion equation, there are generally two bits of information one must have to appropriately describe the mass's motion.

1. The starting position of the mass. 2. The starting direction and magnitude of motion.

Generally, one isn't present without the other. For simplicity, we will consider all displacement below the equilibrium point as and above as .

For upward motion , and for downward motion .

Solution

Multiplying this equation by gives:

The first and the second addends are exact derivatives, so this equation may be integrated to obtain the following relation:

The first addend of this relation is known as the kinetic energy of the mass and the second — as the potential energy of the spring. The above integral represents the energy conservation law. This is also a first order separable differential equation. It may be rewritten as

The integration of this relation gives

Or, finally rearranging the result, substituting , and solving for we obtain