Derive Simple Harmonic Motion

What is Simple Harmonic Motion (SHM)?

The Simple Harmonic Motion (SHM) is defined as the movement of the object in to and fro position along the line. For example; the movement of the pendulum in froth and back position. Simple Harmonic Motion as a Projection of Uniform Circular Motion is defined as the motion of the particle’s projection on any diameter of the reference circle. It is a periodic motion i.e. a body repeats continuously it’s motion on a definite path in a definite time interval, where at equilibrium, moving objects restoring force is directly proportional to the displacement of the object. The Simple Harmonic Motion or Oscillation is the oscillation which is expressed in terms of sine or cosine functions (simple harmonic functions).

Characteristics of Linear Simple Harmonic Motion

The characteristics are as follows;

• The particle motion in a straight line in to and fro position about a fixed point is known as equilibrium position.
• The acceleration or the force is directed always towards the position of equilibrium.
• The restoring acceleration or force acting on the object is directly proportional to the particle’s displacement from the equilibrium position.

Derivation of Simple Harmonic Motion

The elasticity and the inertia are the governing properties of any mechanical systems and is the main cause because of which an oscillating moving body keeps on moving across it’s mean position under the influence of the restoring force (elastic, electrical, gravitational, magnetic, etc.).

Now, at rest if the mass of the body is m which is slightly disrupted by applying the external restoring force. So, according to Hooke’s Law, this restoring external force is directly proportional to the displacement (y) of the particle from it’s mean position.

F ∝ y

F= – k y

where, k represents the proportionality constant or force constant as it relates to the forces from displacement. Now, under the influence of F (restoring force), the body attains the v velocity with a acceleration;

v = dy/dt

and, a = d2y/dt2

According to Newton’s Law of Motion;

F = mass (m) × acceleration (a)

F = m × a

= m (d²y/dt²)

Now, equating both the equations; F= – k y and F = m (d2y/dt2)

– k y = m (d²y/dt²)

d²y/dt² = – (k/m)y

Since k/m is constant therefore, the acceleration (=d²y/dt²) is directly proportional to the y displacement. We can write k/m = w2, so

d²y/dt² = – w²y

d²y/dt² + w²y = 0 (Equation of Simple Harmonic Motion).

We can also write the above equation as;

y = a sin (wt + Φ)

Equations of Simple Harmonic Motion or SHM Equations

Velocity in Simple Harmonic Motion: is the change in the displacement rate with respect to time at that instant.

v = w (a2 – y2)½

Acceleration in Simple Harmonic Motion: is the change in the velocity rate with respect to time at that instant.

a = – w²y

Periodic time of Simple Harmonic Motion: is given by;

T = 2π/w

Equations of displacement of Simple Harmonic Motion in terms of cos: is given by;

x = a cos(wt + Φ)

Total Energy of the particle in SHM (Simple Harmonic Motion): The particle possesses both the kinetic energy (on account of particle velocity) as well as the potential energy (on account of particle displacement from the position of equilibrium).

Potential Energy:

U = (½) mw2y2.

Kinetic Energy:

K = (½) mw2(a² – y²).

Total Energy:

Total Energy (E) =Kinetic Energy(K) + Potential Energy(U)

E = (½) mw²a²

= 2π2mn2a2.

Applications of Linear Simple Harmonic Motion

Few applications are as follows;

• Vertical Oscillation of a column of the liquid in a U-tube.
• Oscillation of the body floating in the liquid medium.
• Oscillation of the ball in the air chamber’s neck.
• Motion of the body dropped in the tunnel across the Earth.

Conclusion

Simple Harmonic Motion or Simple Harmonic Oscillation is the oscillation of the object in to and froth position at equilibrium along the line or circular path. It is the basis of periodic motion for example, Halley’s comet. The relationship between the acceleration, frequency and the displacement is;

a = -4π²n²y.

The restoring force of the object is proportional directly to it’s displacement. The force which bring the vibrating moving body from it’s position of displacement to the position of equilibrium is known as the restoring force.