Periodic Motion

Periodic motion is defined as any motion that repeats at consistent intervals. It is characterized by a specific period, which is the time required to complete one full cycle. This type of motion can be found in various natural and mechanical systems and is essential in many areas, including physics, engineering, and music. Periodic motion is crucial for understanding waves, musical instruments, and the design of clocks and oscillators in electronics. It also plays a vital role in comprehending a range of physical phenomena in nature, from atomic oscillations to the behavior of larger systems.

1.0Definition Of Periodic Motion

Any motion that repeats at regular time intervals is known as periodic motion or harmonic motion. The fixed interval of time after which the motion occurs again is referred to as the time period.

2.0Examples Of Periodic Motion

1. Motion of Planet Around The Sun.
2. Motion Of The Pendulum

3.0Periodic Functions

• Any function that repeats itself at regular intervals of its argument is called a periodic function.

$f(\theta +T)=f(\theta )$

• This indicates that the value of function f remains same when the argument is increased or decreased by an integral multiple of T for all values of .
• A function f satisfying this property is said to be periodic having a period T.
• Trigonometric functions like Sin and Cos are periodic with a period of 2 radians.

$Sin(\theta +2\pi )=Sin\theta$

$Cos(\theta +2\pi )=Cos\theta$

• Checking the Periodicity of Function

$f_1(t)=Sin\frac{2\pi t}{T}$

For checking the periodicity by replacing t by (t+T)

$f_1(t+T)=Sin\frac{2\pi }{T}(t+T)=Sin(\frac{2\pi t}{T}+2\pi )=Sin\frac{2\pi t}{T}=f_1(t)$

$\Rightarrow$Function is Periodic

4.0Representation of Periodic Motion

For representing periodic functions harmonic functions are used. The trigonometric functions of constant amplitude and single frequency are called harmonic functions, among all trigonometric functions only Sin and Cos functions are taken in basic form.

y=A Sin $\theta$=A Sin$\omega$t

y=A Cos $\theta$=A Cos$\omega$t

1. Amplitude: The maximum displacement of particles from the mean is defined as amplitude.
2. Time Period(T):The minimum time after which the particle keeps on repeating its motion is known as time period.

$T=\frac{2\pi }{\omega }T=\frac{1}{n}$, n where n is frequency and is angular frequency.

3. Frequency(f or n): The number of oscillations per second is defined as frequency.

• $f=\frac{1}{T}$, $f=\frac{\omega }{2\pi }$
• SI Unit: Hertz(Hz)1 Hertz = one cycle per second(Cycle is a number not a dimensional quantity)
• Dimension:[$M^0{L}^0{T}^{-1}$]

4. Angular Frequency: The rate of change of phase angle of a particle with respect to time is defined as its angular frequency.

• SI Unit: Radian/Second
• Dimension:$[M^0{L}^0{T}^{-1}]$

5. One Oscillation or Vibration: When a particle goes to one side from mean position and returns back and then it goes to the other side and again returns back to mean position, then this process is known as one oscillation.

5.0Simple Harmonic Motion (SHM) Example Of Periodic Motion

Mean Position: The point at which the restoring force on the particle is zero and potential energy is minimum is known as its mean position.

Restoring Force:

1. The force acting on the particle which tends to bring the particle towards its mean position is known as restoring force.
2. This force is always directed towards the mean position.
3. Restoring force always acts in a direction opposite to that of particle displacement. This displacement is measured from the mean position.
4. It is given by F=-kx and has dimensions $[ML{T}^{-2}]$
5. Direction of displacement and restoring force change at mean position.

Oscillatory Motion

If the body moves back and forth about a fixed point after a certain interval of time.

Linear SHM

F ∝ -x $\Rightarrow$F=-kx

k is restoring force constant

$a=-\frac{k}{m}\Rightarrow \frac{d^{2}x}{d{t}^2}+\frac{k}{m}x=0$

$\frac{d^{2}x}{d{t}^2}+\frac{k}{m}x=0$

It is known as the differential equation of linear SHM.

$x=ASin\omega t\Rightarrow a=-{\omega }^{2}x$

$\omega$ is the angular frequency

${\omega }^2=\frac{k}{m}\Rightarrow \omega =\sqrt{\frac{k}{m}}=\frac{2\pi }{T}\Rightarrow T=\frac{2\pi }{\omega }$

T is the time period and f is the frequency

$T=2\pi \sqrt{\frac{m}{k}}$

$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

6.0Sample Questions On Periodic Motion

Q-1. Which of the following function represents periodic or non-periodic motion?

1. $Sin\omega t+Cos\omega t$
2. $sin\omega t+cos2\omega t+sin4\omega t$

Solution:

(a) $Sin\omega t+Cos\omega t$

$x(t)=sin\omega t+cos\omega t$

$x(t)=\sqrt{2}\left[\mathrm{Sin}\omega t\mathrm{Cos}\frac{\pi }{4}+\mathrm{Cos}\omega t\mathrm{Sin}\frac{\pi }{4}\right]$

$x(t)=\sqrt{2}\mathrm{Sin}\left(\omega t+\frac{\pi }{4}\right)$

$x\left(t+\frac{2\pi }{\omega }\right)=\sqrt{2}\mathrm{Sin}\left[\omega \left(t+\frac{2\pi }{\omega }\right)+\frac{\pi }{4}\right]$

$x\left(t+\frac{2\pi }{\omega }\right)=\sqrt{2}\left(\omega t+2\pi +\frac{\pi }{4}\right)$

$x\left(t+\frac{2\pi }{\omega }\right)=\sqrt{2}\mathrm{Sin}\left(\omega t+\frac{\pi }{4}\right)=x(t)$

(b) $sin\omega t+cos2\omega t+sin4\omega t$

Sint is a periodic function with period = $\frac{2\pi }{\omega }=T$

Cos2t is a periodic function with period = $\frac{2\pi }{2\omega }=\frac{\pi }{\omega }=\frac{T}{2}$

Sin4t is a periodic function with period $=\frac{2\pi }{4\omega }=\frac{\pi }{2\omega }=\frac{T}{4}$

Given function is periodic

Q-2. Are the functions tangent and cotangent periodic, or are they harmonic functions?

Solution: Both $Tan\omega t$ and Cot $\omega$t are periodic functions each with period $T=\frac{\pi }{\omega }$

$\mathrm{Tan}\left[\omega \left(t+\frac{\pi }{\omega }\right)\right]=\mathrm{Tan}(\omega t+\pi )=\mathrm{Tan}\omega t$

$\mathrm{Cot}\left[\omega \left(t+\frac{\pi }{\omega }\right)\right]=\mathrm{Cot}(\omega t+\pi )=\mathrm{Tan}\omega t$

But these functions are not harmonic because they can take any value between 0 and ∞

Q-3. The Earth completes its cycle around its axis in 24 hours, but since there is no oscillatory motion involved, can we say that this rotation is periodic rather than simple harmonic?

Solution: $T=2\pi \sqrt{\frac{l}{g}}$

The girls and the swing together form a pendulum with a certain time period. When the girls stand up, their center of gravity is raised, which reduces the distance between the point of suspension and the center of gravity, effectively shortening the lengthl. As a result, the time period decreases.