Brief Notes On-Time Period And Frequency Of Shm

SHM or Simple Harmonic Motion is a phenomenon where the motion of restoring force is directly proportional to the displacement of the body from its mean position. The restoring force is always directed towards the mean position. 

Let us understand what is a frequency? frequency is the number of oscillations produced by a body per second. The article further elaborates on the frequency formula and frequency practice problems below.

An example of Simple Harmonic Motion in spring; is when a spring is stretched from its resting position, it moves back, compresses and then goes back to its mean position after oscillation creating two extreme positions around the mean position.

Formula for simple harmonic motion:

If restoring force is directly proportional to displacement and applied through opposite direction,

F ∝  x then,

F = –Kx

• F – restoring force

• x – displacement of a body from equilibrium position.

• K – Constant

There are two types of Simple Harmonic Motion they are, 

• Linear SHM (Simple Harmonic Motion)
• Angular SHM (Simple Harmonic Motion)

Linear Simple Harmonic Motion:

When a body moves back and forth from an equilibrium position (fixed point) in a straight line, this motion is called Linear Simple Harmonic Motion 

Example of Linear Simple Harmonic Motion is Spring 

Linear Simple Harmonic Motion is given by,

→    →

F ∝ –x

→    →

a ∝ –x

→    

• F – restoring force

• x – displacement of a body from equilibrium position.

• a – acceleration

Angular Simple Harmonic Motion

When there is oscillation angularly by a body with respect to a fixed axis, this motion is called Angular Simple Harmonic Motion 

Example of Angular Simple Harmonic Motion is  a pendulum

Angular Simple Harmonic Motion is given by, 

Τ ∝ θ or α ∝  θ 

• Τ – Torque

• α – angular acceleration

• θ – angular displacement

Simple Harmonic Motion or SHM Characteristics:

• It is a special case of oscillation along a straight line between the two extreme points (the path of SHM is a constraint)
• Path of the object needs to be a straight line
• There will be a restoring force directed towards equilibrium position (or) mean position
• Mean position in Simple Harmonic Motion is a stable equilibrium

When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time. The string vibrates around an equilibrium position, and one oscillation is completed when the string starts from the initial position, travels to one of the extreme positions, then to the other extreme position, and returns to its initial position.

Simple Harmonic Motion (SHM) is the name of oscillatory motion for a system where Hooke’s law can describe the net force. Such a system is called a simple harmonic oscillator. Suppose Hooke’s law can describe the net force, and there is no damping (by friction or other non-conservative forces). A simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position. 

Periodic motion is that type of motion where an object repeats its motion after a certain period. An example of periodic motion is planets revolving around the sun. This repetitive motion is called periodic motion. Although the revolving of the planets around the sun is a repetitive motion, it doesn’t use restoring force; hence it is not an example of simple harmonic motion.

Oscillation or oscillatory motion is when a body moves back and forth along the same path without changing its shape. A classic example of oscillation or oscillatory motion is the pendulum of a clock. This motion is an example of Simple Harmonic Motion when the oscillation or the displacement is less than 15 degrees in a pendulum. 

What is the time period?

Time period is known as the minimum time after that a body keeps repeating its motion or the shortest time taken to complete one oscillation is also called the time period.

Formula for time period 

T = 2π/ω

The period of a simple harmonic oscillator is given by

T=2π√m/k

and, because 

f = 1/T , the frequency of a simple harmonic oscillator is

f=1/2π√m/k

Note that neither T nor f has any dependence on amplitude.

• T – is period 

• f- is frequency

• m – is mass

• k – is spring constant

What is a frequency?

Frequency is the number of oscillations in a body during the motion in a second.

Frequency Formula,

Frequency = 1/T and, 

Angular Frequency ω = 2πf = 2π/T

Frequency Practice Problems:

1. Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. Describe what happens when the sound decreases to the sound waves in terms of period, amplitude and what is a frequency?

• Frequency and period remain essentially unchanged. Only amplitude decreases as volume decreases

• A babysitter is pushing a child on a swing. At the point where the swing reaches x, maximum distance, where would the corresponding point on a wave of this motion be located?

x is the maximum deformation, which corresponds to the amplitude of the wave. The point on the wave would either be at the very top or the very bottom of the curve.

Conclusion:

We all experience oscillation around us in our daily life. Human beings are surrounded by oscillations such as the beating of the heart, vibration of atoms that make up everything and so on. Simple Harmonic Motion is a very important type of periodic oscillation where acceleration is proportional to displacement from equilibrium in the direction of the equilibrium. In mechanics and physics, Simple Harmonic Motion (sometimes abbreviated SHM) is a special type of periodic motion where the restoring force on the moving object is directly proportional to the magnitude of the object’s displacement and acts towards the object’s equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.