Question

The correct relation between frequency and time period is
A. $T={\displaystyle \frac{1}{f}}$
B. $T=\sqrt{{\displaystyle \frac{1}{f}}}$
C. $T={\displaystyle \frac{1}{2f}}$
D. $T={\displaystyle \frac{2}{f}}$

Answer 1

Hint: Time period is the time taken for one cycle to take complete and frequency is the number of complete cycles that take place in one unit of time.

Complete step by step answer:
Suppose $y=f(t)$ is a function of time such that the function repeats itself after every equal interval of time. For example, $y=\sin (t)$. The individual repetition of the function is called a cycle. The interval of time after which the function repeats itself or the interval of time in which one complete cycle takes place is called the time period of the function. The number of complete cycles of the function that take place in one unit of time is called the frequency of the function. In simple words, frequency is the reciprocal of the time period.
We can understand these two terms better with an example. Let us take the above example, $y=\sin (t)$. If we plot the of y v/s t on the Cartesian plane, we will get the graph the same as $y=\sin (x)$. You may be similar with this graph, so it will be easier to understand. In this graph the values of y repeat after every interval of $2\pi$. Same will happen in the graph of $y=\sin (t)$ . The values of y will repeat after every interval of $2\pi$ seconds (if you take the unit of time as seconds). Therefore, the time period of this function is $T=2\pi \sec$….(1) . The frequency of a time dependent function is the numbers of complete cycles that take place in one unit (1 second) of time. Therefore, if we know the number of complete cycles for a particular amount of time and we divide those numbers of cycles by that time, we will get the value of the frequency. Now, let us do it for the function $y=\sin (t)$. Here we know that 1 complete cycle takes place for $2\pi$ seconds. Therefore, the frequency (f) will be ${\displaystyle \frac{\text{1 cycle}}{\text{2 }\pi \text{ seconds}}}$. A cycle does not have any unit so we write it as $f={\displaystyle \frac{1}{2\pi }}{\mathrm{s}}^{-1}$ .….(2).
From the equations (1) and (2), we get, $f={\displaystyle \frac{1}{2\pi }}={\displaystyle \frac{1}{T}}$
$\Rightarrow T={\displaystyle \frac{1}{f}}$
Hence, the correct option is (a) $T={\displaystyle \frac{1}{f}}$.

Note: Do not get confused between frequency ($f$) and angular frequency($\omega$). Both the terms are similar with little difference. $f={\displaystyle \frac{1}{T}}$ and $\omega ={\displaystyle \frac{1}{2\pi T}}$.
$\Rightarrow f=2\pi \omega$.