Periodic Function: Examples, Formula, and Derivation

Ans. A periodic function, also known as a cyclic function or a periodic waveform, is a function whose values repeat at regular intervals or periods. The repeated part of a function or waveform is referred to as a Cycle. Periodic functions include trigonometric functions, which occur at radian intervals.

Ans. Some examples of periodic functions are

• Sine wave
• Triangular wave
• Square wave
• Sawtooth wave

Ans. A function f is considered to be periodic, if

f (x + P) = f (x)

Where P is a non-zero constant, known as the period of the function.

Ans. (a) sin ωt - cos ωt

⇒ sin ωt - cos ωt = √2 [sin ωt (1/√2) - cos ωt (1/√2)]

Since cos π/4 = sin π/4 = 1/√2

⇒ sin ωt - cos ωt = √2 [sin ωt cos π/4 - cos ωt sin π/4]

Using the formula sin (A - B) = sin A cos B - cos A sin B, we get

sin ωt - cos ωt = √2 [sin ωt - π/4]

Comparing it with the simplest equation of periodic motion 

y = A sin ωt ± Φ

Where

• A is the maximum displacement of the particle
• Φ is the initial phase

Hence the above equation represents periodic motion.

Now, let T be the time period to complete one revolution (2π), then

ωT = 2π

⇒ T = 2π/ω

(b) log (2ωt)

The function log (2ωt) increases monotonically with time t. It, therefore never repeats its volume. Hence it is a non-periodic function.

As t → ∞, log (2ωt) → ∞

Ans. A motion that repeats itself after equal intervals of time is known as periodic motion.

Ans. The correct answer is a. True

Explanation: Yes, a motion can be periodic but not oscillatory. For example, uniform circular motion is periodic but not considered an oscillatory motion.

Ans. The given periodic function is 

f(ax + b) = cos(5x + 4)

The coefficient of x is, a = 5

If the function f(x) is periodic with time T, then f(ax + b) is periodic with time T/|a|

The period of cos x is, T = 2π

Hence the period of cos(5x + 4) is 2π/5

Ans. The amplitude of a periodic function is the distance between the maximum and minimum values of the function.

Ans. The frequency of a periodic function is the number of times the function repeats itself in one period.

Ans. (a) x = a sin ωt + b cos ωt

As x is a function of time i.e. x(t) = a sin ωt + b cos ωt

We know 

sin (θ + 2nπ) = sin θ and

cos (θ + 2nπ) = cos θ

We can write,

x(t) = a sin (ωt + 2nπ) + b cos (ωt + 2nπ)

⇒ x(t) = a sin ω(t + 2nπ/ω) + b cos ω(t + 2nπ/ω)

⇒ x(t) = x(t + 2nπ/ω)

The value of x at t and t + 2nπ/ω is the same. Therefore x is a periodic function with a time period of 2π/ω.

(b) x = sin ωt + sin 2ωt + sin 3ωt

⇒ x(t) = sin ωt + sin 2ωt + sin 3ωt

Also, x(t + nT) = sin ω(t + nT) + sin 2ω(t + nT) + sin 3ω(t + nT)

Let T = 2π/ω

⇒ x(t + nT) = sin ω(t + n 2π/ω) + sin 2ω(t + n 2π/ω) + sin 3ω(t + n 2π/ω)

⇒ x(t + nT) = sin (ωt + 2nπ) + sin (2ωt + 4nπ) + sin (3ωt + 6nπ)

⇒ x(t + nT) = sin ωt + sin 2ωt + sin 3ωt

⇒ x(t + nT) = x(t)

So x is a periodic function with a time period T = 2π/ω

Ans. The correct answer is a. n/2

Explanation: The energy of an oscillation becomes kinetic and potential twice in one oscillation. Kinetic energy frequency = 2n.

Ans. The following are the differences between periodic and simple harmonic motion

Ans. Periodic motion is defined as motion that repeats itself at equal time intervals. The time interval is referred to as the time period of periodic motion. Examples of periodic motion are

• the motion of a wall clock's pendulum
• the motion of planets around the sun
• the motion of a clock's hands

Oscillatory or vibratory motion is defined as a body moving back and forth repeatedly about a mean position. Examples of oscillatory motion are

• the vibrations of the string of a sitar
• the motion of the pendulum of a clock

Every oscillatory motion is periodic. A periodic motion may be oscillatory or not. The motion of planets around the Sun, for example, is always periodic but not oscillatory. The motion of a clock's pendulum is both periodic and oscillatory.

Ans. Motion is defined as the ability of a body to change its position within a certain time frame. They can be further classified based on the characteristics of the motion. Some of the most common types of motion are:

• Oscillating motion
• Reciprocating motion
• Rotary motion
• Linear motion

On the other hand, Simple Harmonic Motion (SHM) is a type of oscillating motion. The total force exerted on an object in this type of motion is actually restoring force. SHM can thus be represented as periodic motion. The object tends to move to and fro along a fixed point in this motion. the motion of the clock pendulum is one of the best examples of simple harmonic motion.

Ans. The correct answer is b. Zero

Explanation: The momentum imparted to the sphere by the moving bob is 0 since the velocity of the bob at the end of the arc is zero.