Periodic motion  (Page 2/2)

Elements of periodic motion

Here, we describe certain important attributes of periodic motion, which are extensively used to describe a periodic motion. Though, it is expected that readers are already familiar with these terms, but we present the same for the sake of completeness.

1: Time period (T) : It is the time after which a periodic motion (i.e. the pattern of motion) repeats itself. Its dimensional formula is [T] and unit is “second” in SI unit.

2: Frequency (n, ν) : It is the number of times the unit of periodic motion is repeated in unit time. In SI system, its unit is $s^{-1}$ , which is known as "Hertz" or "Hz" in short. An equivalent name is cycles per second (cps). Its dimensional formula is [ $T^{-1}$ ]. Time period and frequency are inverse to each other :

$$T=\frac{1}{\nu }$$

3: Angular frequency (ω) : It is the product of “2π” and frequency “ν”. In the case of rotational motion, angular frequency is equal to the angle (radian) described per unit time (second) and is equal to the magnitude of average angular velocity.

$$\omega =2\pi \nu$$

The unit of angular frequency is radian/s. In general, angular frequency and angular velocity are referred in equivalent manner. However, we should emphasize that we refer only the magnitude of angular velocity, when quoted to mean frequency. Also,

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

Since “2π” is a constant, the dimensional formula of angular frequency is same as that of frequency i.e. [ $T^{-1}$ ].

4: Displacement (x or y) : It is equal to change in the physical quantity in periodic motion. This physical quantity can be any thing like displacement, electric current, pressure etc. The unit of displacement obviously depends on the physical quantity under consideration.

Period of periodic motion

There are many different periodic functions. In our course, however, we shall be dealing mostly with trigonometric functions. Some important results about period are useful in finding period of a given function.

1: All trigonometric functions are periodic.

2: The periods of sine, cosine, secant and cosecant functions are “2π”, whereas periods of tangent and cotangent functions are “π”.

3: If "k","a" and "b" are positive real values and “T” be the period of periodic function “f(x)”, then :

• "kf(x)" is periodic with period “T”.
• "f(x+b)" is periodic with period “T”.
• "f(x) + a" is periodic with period “T”.
• "f(ax±b)" is periodic with a period “T/|a|”.

4: If “a” and “b” are non-zero real number and functions g(x) and h(x) are periodic functions having periods, “ $T_1$ ” and “ $T_2$ ” , then function $f\left(x\right)=ag\left(x\right)\pm bh\left(x\right)$ is also a periodic function. The period of f(x) is LCM of “ $T_1$ ” and “ $T_2$ ”.

Examples

Problem 2: Find the time period of the motion, whose displacement is given by :

$$x=\cos \omega t$$

Solution : We know that period of cosine function is “2π”. We also know that if “T” is the period of “f(t)”, then period of the function "f(at±b)" is “T/|a|”. Following this rule, time period of the given cosine function is :

$$\Rightarrow T=\frac{2\pi }{\omega }$$

Note that this expression is same that forms the basis of definition of angular frequency.

Problem 3: Find the time period of the motion, whose displacement is given by :

$$x=2\cos \left(-3\pi t+5\right)$$

Solution : We know that period of cosine function is “2π”. If “T” is the time period of f(t), then period of kf(t) is also “T”. Thus, coefficient “2” of trigonometric term has no effect on the period. We also know that if “T” is the period of “f(t)”, then period of the function "f(at±b)" is “T/|a|”. Following this rule, time period of the given cosine function is :

$$\Rightarrow T=\frac{2\pi }{3\pi }=\frac{2}{3}$$

Problem 4: Find the time period of the motion, whose displacement is given by :

$$x=\sin \omega t+\sin 2\omega t+\sin 3\omega t$$

Solution : We know that period of sine function is “2π”. We also know that if “T” is the period of “f(t)”, then period of the function "f(at±b)" is “T/|a|”. Following this rule, time periods of individual sine functions are :

$$\Rightarrow \text{Time period of "sin ωt"},T_1=\frac{T}{\omega }=\frac{2\pi }{\omega }$$

$$\Rightarrow \text{Time period of "sin 2ωt"},T_2=\frac{T}{2\omega }=\frac{2\pi }{2\omega }=\frac{\pi }{\omega }$$

$$\Rightarrow \text{Time period of "sin 3ωt"},T_3=\frac{T}{3\omega }=\frac{2\pi }{3\omega }=\frac{2\pi }{3\omega }$$

Applying LCM rule, we can find the period of combination. Now, LCM of fraction is obtained as :

$$\Rightarrow T=\frac{\text{LCM of numerators}}{\text{HCF of denominators}}=\frac{\text{LCM of “2π”, “π” and “2π”}}{\text{HCF of “ω”, “ω” and “3ω”}}$$

$$\Rightarrow T=\frac{2\pi }{\omega }$$

It is intuitive to understand that the frequencies of three functions are in the proportion 1:2:3. Their time periods are in inverse proportions 3:2:1. It means that by the time first function completes a cycle, second function completes two cycles and third function completes three cycles. This means that the period of first function encompasses the periods of remaining two functions. As such, time period of composite function is equal to time period of first function.