Question

Determine whether the given function is periodic. If so, find the period.
1-\[\sin wt-\cos wt\]
2- \[\log (2wt)\]

Answer 1

Hint: In general, the trigonometric functions like sine and cosine are periodic which is evident from the nature of their graph. But it is not necessary that the sum of two periodic functions is also a periodic function. Any function will represent a periodic function if it can be written in the form of sin(ωt+ϕ) where its period is\[\dfrac{2\pi }{w}\]

Complete step by step answer:
1- the given function is composed of sin and cosine. We know both sine and cosine are periodic functions, thus this will also be periodic. Let us verify it mathematically,
\[\begin{align}
  & \Rightarrow \sin wt-\cos wt \\
 & =\sqrt{2}\{\dfrac{1}{\sqrt{2}}\sin wt-\dfrac{1}{\sqrt{2}}\cos wt\} \\
 & =\sqrt{2}\{\sin 45\sin wt-\cos 45\cos wt\} \\
 & =\sqrt{2}\sin (wt-45) \\
\end{align}\]
This function represents a periodic function as it can be written in the form: a sin(ωt+ϕ) Its period is\[\dfrac{2\pi }{w}\]

2- \[\log (2wt)\], we know logarithmic functions are not periodic functions, they either increase or decrease. logarithmic functions do not repeat themselves. Therefore, it is a non-periodic motion.

Additional information- Frequency is the number of complete cycles per second for an oscillating body. It is the characteristic of the source and as the wave propagates through the medium it does not change.

Note- When a body vibrates then it undergoes to and fro motion and this is a type of motion is called periodic motion. As the name suggests it has a characteristic period of executing the motion back again and again. Periodic motion has a time period which is defined as the time taken by the body to return to its initial position.