Simple Harmonic Motion

SHM stands for Simple Harmonic Motion, which is described as a motion in which the displacement of the body from its mean position is directly proportional to the restoring force. This restoring force always moves in the direction of the mean position. The acceleration of a particle in simple harmonic motion is given by

a(t)=-ω²x(t) 

Here is the particle’s angular velocity.

Simple harmonic motion is a kind one oscillatory motion in which the acceleration of the particle is directly proportional to its displacement from the mean location at any point. It’s a type of oscillatory motion but a little different.

Simple Harmonic Motions (SHM) are oscillatory and periodic in nature, but not all oscillatory motions are SHM. The harmonic motion of all oscillatory motions, the most important of which is simple harmonic motion, is known as oscillatory motion (SHM).

The velocity, acceleration, displacement, and force in this kind of oscillatory motion fluctuate (with respect to time) in a way that may be characterised by either sine (or) cosine functions, generally known as sinusoids.

Periodic Motion

It is defined as a motion that repeats itself at regular intervals of time. A tuning fork or the motion of a pendulum are examples of periodic motion; if you analyse the motion, you’ll notice that the pendulum only passes through the mean position after a specific amount of time. The type of motion that is described above can alternatively be classified as oscillatory. An oscillatory motion occurs when the body moves back and forth around a fixed point. As a result, an oscillatory motion may or may not be periodic.

Oscillation Motion

The to and fro motion of an object from its mean location is known as oscillatory motion. The ideal situation is for the object to remain in oscillatory motion in the absence of friction indefinitely, since this cannot be achieved in real life scenarios, so the object comes to rest after a period of time.

Simple Harmonic Motion

It’s a type of oscillation with a straight line connecting the two extreme locations (the path of SHM is a constraint). The path of the object must be following a straight line. A restoring force will be directed towards the equilibrium (or mean) position. In simple harmonic motion, the mean position is a stable equilibrium.

Difference between Oscillatory and Periodic Motion:

Periodic motion is described as movement that repeats itself at regular intervals. The periodic motion’s time period is defined as a definite interval of time. The movements of the hands of a clock, the motion of planets around the sun, and so on are examples of periodic motion.

The to and fro motion of the body around its fixed position is referred to as oscillatory motion. Periodic motion includes oscillatory motion

Types of Simple Harmonic Motion

Linear Simple Harmonic motion:

The simplest form of oscillatory motion is linear S.H.M, in which a body is displaced from its mean position and oscillates ‘to and fro’ about its mean position, with the restoring force always oriented towards the mean position and its size directly proportional to the displacement.

Example: 

• The oscillations of a simple pendulum is a type of SHM.
• Alternating current

Angular Simple Harmonic Motion:

Angular oscillations can be produced by a body that is free to rotate around an axis. A photo frame or a calendar, for example, could be hung from a nail on the wall. It creates angular oscillations when pushed slightly from its mean position and then released.

Time period and frequency of Simple Harmonic Motion:

The time period (or) the smallest time taken to complete one oscillation is also described as the minimal duration after which the particle continues to repeat its motion.

T=2π/ω

The frequency is defined as the number of oscillations per second.

Frequency=1/T

ω=2πf=2π/T

Phase:

The phase of a vibrating particle at any given time is the state of the vibrating (or oscillating) particle in terms of displacement and vibration direction at that time.

The expression for a particle’s location as a function of time.

x=A sin sin (ωt+ø)

The phase angle at time t = 0 is known as the initial phase, where ωt+is the particle’s phase.

Total energy of a particle undergoing SHM

The total energy of a block and spring system is proportional to the square of the amplitude and is equal to the sum of the potential energy stored in the spring plus the kinetic energy of the block.

E=1/2mω²(A²-x²)+1/2mω²x²

E=1/2mω²A²

As a result, in SHM, the particle’s total energy is constant and independent of its instantaneous movement.

Conclusion:

The one-dimensional projection of uniform circular motion can be thought of as simple harmonic motion. If an object moves with angular speed around a circle of radius r centred at the xy-origin, plane’s its motion along each coordinate is a simple harmonic motion with amplitude r and angular frequency.