Q.Which of the following functions of time represent

Concept: Frequency and Period

A function is periodic if it repeats after a fixed time interval $T$, i.e., $f(t+T) = f(t)$ for all $t$. The period is the least positive $T$ satisfying this.

(i) $\sin\omega t + \cos\omega t$

Rewrite: $\sqrt{2}\sin(\omega t + \pi/4)$. This is a single sinusoid with angular frequency $\omega$.

Period $T = \frac{2\pi}{\omega}$.

Periodic.

(ii) $\sin\omega t + \cos 2\omega t + \sin 4\omega t$

Angular frequencies: $\omega$, $2\omega$, $4\omega$. Their periods are $T_1 = 2\pi/\omega$, $T_2 = \pi/\omega$, $T_3 = \pi/(2\omega)$. The fundamental period is the LCM of these: $2\pi/\omega$.

Periodic with $T = \frac{2\pi}{\omega}$.

(iii) $e^{-\omega t}$ …

Let’s understand what “periodic motion” really means. A function $f(t)$ is periodic if there exists a positive constant $T$ such that $f(t+T) = f(t)$ for every $t$. The smallest such $T$ is called the fundamental period. Intuitively, the graph repeats itself exactly after every interval of length $T$.

For combinations of sine and cosine, the period is the least common multiple (LCM) of their individual periods. For exponential and logarithmic functions, we need to check if they ever repeat — they usually don’t, because they are strictly monotonic (always increasing or decreasing).

Now let’s examine each case.

1. (i) $\sin\omega t + \cos\omega t$

First, note that $\sin\omega t$ has period $T_1 = 2\pi/\omega$, and $\cos\omega t$ also has period $T_2 = 2\pi/\omega$. Since both have the same period, their sum will also have that period. But is it the fundamental period? Let’s check.

We can rewrite the sum as a single sine wave:

This is a sine function with angular frequency $\omega$, so its period is indeed $T = 2\pi/\omega$.

Thus, (i) is periodic with period $\boxed{2\pi/\omega}$.

2. (ii) $\sin\omega t + \cos 2\omega t + \sin 4\omega t$

Find the individual periods:

• $\sin\omega t$: $T_1 = 2\pi/\omega$
• $\cos 2\omega t$: $T_2 = 2\pi/(2\omega) = \pi/\omega$
• $\sin 4\omega t$: $T_3 = 2\pi/(4\omega) = \pi/(2\omega)$

The sum is periodic if there exists a common $T$ that is an integer multiple of each of these. That common period is the LCM of $T_1, T_2, T_3$.

Write them as:

The LCM of $2\pi/\omega$, $\pi/\omega$, and $\pi/(2\omega)$ is $2\pi/\omega$ (since $2\pi/\omega$ is a multiple of each: $2\pi/\omega = 2 \times \pi/\omega = 4 \times \pi/(2\omega)$). …

Step 1: A function $f(t)$ is periodic if there is a smallest $T>0$ with $f(t+T)=f(t)$ for every $t$; for a sum of sinusoids the period is the LCM of the individual periods, since the whole sum returns to its starting value only once every component has completed a whole number of cycles.

Step 2: (i) $\sin\omega t+\cos\omega t=\sqrt2\sin(\omega t+\pi/4)$ is a single sinusoid of angular frequency $\omega$, so it is periodic with $T=2\pi/\omega$.

Step 3: (ii) The three terms $\sin\omega t,\cos2\omega t,\sin4\omega t$ have periods $2\pi/\omega,\pi/\omega,\pi/(2\omega)$; their LCM is $2\pi/\omega$, so the sum is periodic with $T=2\pi/\omega$. …