All About The Time Period of SHM

It is very common to see oscillations as it is happening all around us, from the vibration of the atoms to the beating of a human heart. Simple Harmonic Motion is a periodic oscillation. However, you can measure its period( the time it consumes for one oscillation); the formula of time period is T =1/f, where t refers to time period, and f refers to the frequency and based on it, you can find out its frequency ( the number of oscillations per unit of time and the inverse of the time period).

What is Simple Harmonic Motion?

Simple Harmonic Motion is the motion where the restoring force is directly proportional to the object’s displacement from its mean position. The direction of this restoring force always stands out towards the mean position. The acceleration of a particle executing the simple harmonic motion is given out by a(t)= -ω2 x(t), where the ω is the angular velocity of the particle.

All the simple harmonic motions are oscillatory, but not all oscillatory motions need to be SHM (Simple Harmonic Motion). Oscillatory motion is the harmonic motion of all the oscillatory motion, where Simple Harmonic Motion is the most important one. And in such a type of oscillatory motion, velocity, acceleration, displacement, and force can be defined by either sine Or the cosine, which are collectively called sinusoids. 

The study of the simple harmonic motion is very useful and forms an important tool in understanding the characteristics of sound waves, light waves, and back and forth. Any oscillation motion that is not a simple harmonic can be expressed as a superposition of several harmonic movements from different frequencies.

Linear Simple Harmonic Motion 

When a particle moves here and there about the fixed point (called the equilibrium position) along with a straight line, the movement is called a simple linear harmonic movement.

For example, A spring-mass system.

What is the Frequency and time period? 

The time period (symbolised by ‘T”) is the time taken for one complete vibration cycle to pass a certain point. Because the wave frequency increases, the wave period is reduced. The unit for the time period is ‘second.’ Frequency and time period are in a reciprocal relationship. The formula of time period is: T = 1 / F or as: F = 1 / T.

Because simple harmonic movements are periodic oscillations, we can measure the period (the time needed for one oscillation) and determine the Frequency (the amount of oscillation per unit of time or the opposite of the period).

The two most common trials that show this are:

1. Pendulum 

– where the mass m is sticking to the end of the long pendulum, it will oscillate with the period (T). It is explained by: t = 2π√ (l / g), where g is acceleration due to gravity.

1. The mass in the spring 

-where the mass m is sticking to the spring with a spring constant k will oscillate with the period (T). Explained by: T = 2π√ (m / k).

By determining the duration of one complete oscillation, we can determine the period and frequency. In the case of a simple harmonic oscillator, the period is independent of the amplitude.

From the definition, acceleration, a, from an object in a simple harmonic motion is proportional to its displacement, x:

a = -ω2x

Where ω is the Frequency of angles and can be determined by knowing the period (Ω = 2π / t) or Frequency (ω = 2πf), given that speed (V) is the derivative time of distance and acceleration is time derivative speed, that when starting from amplitude (a), the solution follows the sinusoidal function from form x = a cos (ωt).

Conclusion: 

In the above article, we learn about SHM and the time period of SHM, the formula of the time period, and much more. SHM is just an acceleration of any particle that is directly proportional to the object’s displacement to its mean position. We also learn in detail about periodic and oscillatory motion and the frequency and time period. So I hope the above article helps you a lot to have a clear and deep understanding of the time period of the SHM formula.