Simple Harmonic Motion

When we study about harmonic motions, first we will just examine special motions in Nature. And you may have seen this kind of motion many times in your real life which is called Simple Harmonic Motion.

What is special about Simple Harmonic motion is that, we can completely explain its motion using basic Physics concepts. We can get the complete solution of Equations of motion.

So, instead of talking we can analyze it.

The simplest example is spring with mass m.

We know that the spring will oscillate back and forth if you give a little force to the mass m. On analyzing carefully we get that when the spring is elongated, the force tries to compress and when it is compressed the force tries to elongate it.

This shows that the force always tries to make the spring in constant length. It doesn’t like to change its length. At certain length, the spring doesn’t do anything i.e. neither compressing nor elongating force. This position is called the Equilibrium Position. And the length of the spring at the equilibrium position is commonly known as the length of the spring.

And the elongation or the compression length with its direction could be written as the displacement vector x which is the displacement made by the mass m from the equilibrium position.  

So, from the Newton’s second law of motion it is known that the force acting on the mass is

m d^2 x/ dt^2

When we need to study about the motion of the spring, we will just care about the force acting on the mass.

When I said force, don’t think about what kind of force would be there because without knowing the force itself we can determine its motion.

We don’t need to care whether the forces are gravitational or electrostatic or the electromagnetic or the combination of all these. For our task we just need the total effect.

And from the output it can be deduced that the force should be proportional to the elongation or compression length and should be directed opposite towards the displacement.

Hence it is written as

      

                   m (d^2 x⃑   / dt^2)              α       – x⃑

                    m (d^2 x⃑   /  dt^2)              =       – k  x⃑

   ⃑    used to denote that it is a vector  

the value “k” is the proportionality constant. We need to analyze it.  

The above equation in words follows as,

In Nature, if any object follows an absolute rule that its second order rate of change of position is proportional to the position of the object itself, then it follows the harmonic motion. Provided its mass is not changing with respect to time.

But you may instantaneously ask the question why it is not proportional to square, cube or any other power of the elongation or compression length.   

         

 The answer is yes. The force does really depends on the other powers of x. But mostly we observe the first power in these motions. And moreover, only for the first power of “x” it is easy to solve for the equations of motion.

The difficulty of solving the equations for higher powers of “x” is too much difficult and it will be no more a simple harmonic motion.

That doesn’t mean we shouldn’t think about other powers. But we will do it by step by step.

Therefore in our first step we are confined only to the first power of “x”.

and the differential equation for that motion is given by,

                m d^2 x⃑ / dt^2  +    k  x⃑    =   0

Since “m” is a constant

                d^2 x⃑ / dt^2       +    (k/m)  x⃑    =   0

or              d^2 x⃑ / dt^2       +    A  x⃑    =   0

where A = k/m

This is just a second order differential equation. Now it is a mathematical problem.