Animated Solution for Physics - Oscillations: The function ${\sin }^2(\omega t)$ represents

When we encounter a function like $y={\sin }^2(\omega t)$, our first instinct might be to classify it as Simple Harmonic Motion (SHM) because it involves a sine function. However, physics demands rigorous mathematical proof. Let's dive into the calculus to uncover the true nature of this motion.

The golden rule for any motion to be classified as SHM is that its acceleration must be directly proportional to the negative of its displacement from the mean position. Mathematically, this is expressed as:

To find the acceleration, we need to differentiate our displacement function twice with respect to time. Let's start with velocity:

Using the chain rule, we get:

Now, let's differentiate the velocity to find the acceleration:

Now, let's compare our acceleration with our original displacement. We have $a=2{\omega }^2\cos (2\omega t)$ and $y={\sin }^2(\omega t)$. It is glaringly obvious that $a$ is not proportional to $-y$. Therefore, the motion represented by $y={\sin }^2(\omega t)$ is not Simple Harmonic Motion.

Even though it's not SHM, the function is clearly periodic. It repeats itself over regular intervals. To find its time period, we can use a handy trigonometric identity to simplify the function:

Applying this to our function, we get:

The periodic nature of this function is entirely governed by the $\cos (2\omega t)$ term. The angular frequency of this term is ${\omega }^{\prime }=2\omega$.

The time period $T$ of any periodic function is given by $2\pi$ divided by its angular frequency. Therefore:

So, the function represents a periodic motion that is not simple harmonic, and it has a time period of $\frac{\pi }{\omega }$.