Animated Solution for Physics - Oscillations: A simple harmonic motion is represented by $y=5(\sin 3\pi t+\sqrt{3}\cos 3\pi t)$ cm. The amplitude and time period of the motion are

Imagine two waves dancing together. One is a sine wave, starting from zero, and the other is a cosine wave, starting from its peak. When these two waves overlap, they don't just create chaos; they combine to form a single, beautiful, new wave. This phenomenon is known as the superposition of simple harmonic motions.

In our problem, we are given the equation:

At first glance, this looks like a complicated mix of two different oscillations. But physics and mathematics give us a powerful tool to merge them into one elegant equation.

To combine a sine and a cosine term of the same frequency, we use a clever algebraic trick. We look at the coefficients of the sine and cosine terms inside the bracket, which are $1$ and $\sqrt{3}$.

We imagine these coefficients as the base and perpendicular of a right-angled triangle. The hypotenuse of this triangle would be:

So, we multiply and divide our expression by $2$:

Now, we recognize that the fractions inside the bracket correspond to exact trigonometric values. Specifically, $\cos \left(\frac{\pi }{3}\right)=\frac{1}{2}$ and $\sin \left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}$. Substituting these into our equation gives:

This expression perfectly matches the standard trigonometric identity for the sine of a sum: $\sin (A+B)=\sin A\cos B+\cos A\sin B$. Applying this identity, our equation simplifies dramatically:

We have successfully transformed the complex superposition into the standard equation of a Simple Harmonic Motion:

By comparing our simplified equation with the standard form, we can directly read off the physical parameters of the motion. The amplitude $A$ is the coefficient outside the sine function:

The angular frequency $\omega$ is the coefficient of $t$:

Finally, the time period $T$, which is the time taken to complete one full oscillation, is given by the relation $T=\frac{2\pi }{\omega }$. Substituting our value of $\omega$:

And there we have it! By using a simple trigonometric substitution, we unlocked the true amplitude and time period of the combined motion.