What is the Mathematical Analysis of Simple Harmonic Motion?

Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.

What is Differential Equation for SHM?

Let x be the displacement from equilibrium position of a particle executing SHM.
Then the velocity of the particle is, v = \(\frac{d x}{d t}\).
The acceleration of the particle (i.e., the rate of change of velocity),
a = \(\frac{d v}{d t}\) = \(\frac{d}{d t}\left(\frac{d x}{d t}\right)\) = \(\frac{d^2 x}{d t^2}\)
So, the equation of SHM can be written as
\(\frac{d^2 x}{d t^2}\) = -ω²x or, \(\frac{d^2 x}{d t^2}\) + ω²x = 0 ….. (1)
Equation (1) is called the differential equation of SHM.

Solution of the differential equation: Let x = e^pt be a solution for equation (1).
∴ \(\frac{d x}{d t}\) = pe^pt and \(\frac{d^2 x}{d t^2}\) = p²e^pt
Putting the values obtained above in equation (1) we have,
p²e^pt + ω²e^pt = o
or, (p² + ω²)e^pt = 0
or, p² + ω² = 0 [∵ e^pt ≠ 0]
or, p² = -ω² or, p = ± iω [Here, i = \(\sqrt{-1}\)]
So, the general solution of equation (1) is
x = A’sinωt + B’cosωt ………. (2)
where A’ and B’ are integration constants.

Inserting e^mx in the equation \(\frac{d^2 y}{d x^2}\) + c₁\(\frac{d y}{d x}\) + c₂y = 0, if we get m = a’ ± ib’, then the general solution of the equation is y = e^a’x(A”sinb’x + B”cosb’x).

Let us put A’ = Acosα and B’ = Asinα in equation (2).
Then, A = \(\sqrt{A^{\prime 2}+B^{\prime 2}}\) and α = tan^-1\(\frac{B^{\prime}}{A^{\prime}}\)
Thus, x = A (sinωtcosα + cosωtsinα)
or, x = Asin (ωt + α) …….. (3)

This represents the general equation of simple harmonic motion expressing the displacement of the particle.

Special cases:

i) If A’ = A and B’ = 0, i.e., a = 0, we have from equation (2) or (3),
x = Asinωt ……….. (4)
From this equation, it is seen that, at t = 0, x = 0,
i.e., initially the particle is at its position of equilibrium.

ii) If A’ = 0 and B’ = A, i.e., α = \(\frac{\pi}{2}\), we have from equation (2) or (3),
x = Acosωt ………….. (5)

From this equation, it is seen that, when t = 0, x = A, which is the maximum value of displacement of the particle.

So, if a particle executing SHM starts its motion from its equilibrium position, its displacement is expressed as a sine function. If it starts its motion from one end of its path, its displacement is expressed as a cosine function.

If the particle executing SHM starts its motion from any other point of its path, then equation (3) is directly used.

It is to be noted that, like equations (1) and (2) in Section 1.1.5, equations (3), (4) and (5) are also representative equations of SHM.