Simple Harmonic Motion

Simple harmonic motion involves a simple repetitive motion of back and forth, a central position, and the furthermost displacement on either side of the central position is the same. Each vibration is completed in the same time period. The force of motion is directed towards the central position at all times during the motion and the force is directly proportional to the distance from the central position.

F = -kx 

Where F is the force, x is the displacement, and k is restoring/spring constant. This is known as Hooke’s law.

Key terms:

• Frequency: The number of oscillations completed per unit of time is the frequency of the simple harmonic motion being performed by the particle. The SI unit of frequency is Hertz. 

• Amplitude: The magnitude of maximum displacement of a particle from its mean position during simple harmonic motion is called its amplitude. 

• Time period: The time in which the particle completes one oscillation is called its time period. 

• Mean position of equilibrium: The position of the particle during the motion when the net force acting on it is zero.

Frequency and period of simple harmonic motion

The period or the time taken to complete one oscillation and the frequency or the number of oscillations per unit time of simple harmonic motion can be calculated because it is a periodic motion. Two experiments can be used for this purpose:

1. Pendulum: A mass m is attached to a pendulum of length l. It will oscillate in a period T. the following formula describes the motion

T = 2π√(l/g), where g is the gravitational acceleration 

1. Mass on a spring: A mass m is attached to a spring with a spring constant k. The period t is described by the following formula:

T = 2π√(m/k)

In the case of the pendulum, the time period is independent of the mass whereas in the mass spring the time period depends on the mass.

By calculating the time period for one oscillation the frequency can be easily calculated. The period of a simple harmonic motion body is not dependent on its amplitude.

When we study the characteristics of simple harmonic motion we can arrive at the conclusion that the acceleration of the oscillating body is directly proportional to its displacement. This relationship can be described by the following formula:

a = -ω2x

Where ω is the angular frequency and can be determined by calculating the period or the frequency. If we remember that velocity is the time derivative of displacement and acceleration is the time derivative of velocity it can be concluded that the solution follows a sinusoidal function described by the following form:

x = A cos(ωt)

Simple harmonic motion also demonstrates the conversion of potential energy into kinetic energy and vice versa. When the mass is at the position of maximum displacement the velocity is zero and the potential energy is maximum. But when the mass is at the equilibrium position during the oscillation the velocity and kinetic energy are maximum.

Types of SHM (Simple Harmonic Motion)

• Linear simple harmonic motion: When a mass moves back and forth in a straight line around an equilibrium position it is known as a linear simple harmonic motion. An essential condition followed by this type of motion is that the force is indirectly proportional to the displacement represented by:

F ∝ – X

Where  F is the force and X is the displacement from the position of equilibrium.

• Angular simple harmonic motion: A body that demonstrates simple harmonic motion along with a change in its angular position is said to be performing angular simple harmonic motion. This type of motion happens when the angular acceleration or the torque is directly proportional to the displacement of the body from its position of equilibrium. That is:

T ∝ -θ

Phase in Simple Harmonic Motion

The phase of a vibrating particle is its position in relation to its displacement and direction of displacement. It can be described the following formula:

x = A sin (ωt + Φ)

In this formula (ωt + Φ) is the phase of the oscillating particle. At t = 0 the phase is called the initial phase.

Phase difference

The difference between the phase angles of two particles describing a simple harmonic motion with respect to the mean position is called phase difference. When the phase difference is an even multiple of π, the particles are said to be in the same phase.

ΔΦ = nπ where n = 0, 1, 2, 3, . . .

If the phase difference is an odd multiple of π, the particles are said to be the opposite phase.

ΔΦ = (2n + 1) π where n = 0, 1, 2, 3, . . .

Examples of simple harmonic oscillators

• Uniform circular motion: an object moving with angular velocity around a fixed point along a path describing a circle then the object is said to be going in a simple harmonic motion with its amplitude being equal to the radius of the circle path. 

• Simple pendulum: The motion of a simple pendulum is simple harmonic motion. 

• Oscillatory motion: The two and fro motion of a body described about a certain point is known as oscillatory motion and is an example of simple harmonic motion. 

• Scotch yoke mechanism: In this type of mechanism rotational motion and linear reciprocating motion are interchanged continuously.

Conclusion 

Simple harmonic motion is a part of mechanics in physics. It is a special type of periodic motion with particular conditions and characteristics. In simple harmonic motion, the acceleration is proportional to the displacement of the particle from the position of equilibrium. This type of motion is oscillatory and oscillations happen all around us all the time. The beating of the heat, the flapping of a hummingbird’s wings, even the vibrations of the atoms that make all matter, oscillations are everywhere. So it makes sense to understand how this type of motion is described by bodies and how objects behave under this motion.