Derivation of SHM equation

Oscillations & Waves-SHM-Equation of SHM_Physics

Introduction.

All the Simple Harmonic Motions are periodic motion. In SHM, the object oscillates from its mean position to the extreme position and back to the mean position. During this whole process of oscillation, the restoring force is experienced by the oscillating object, which is directly proportional to the magnitude of the displacement of an object from its mean position but is in the opposite direction to the displacement. It can be written as, 

F α –x 

The real-life examples of SHM are cradle, swing, pendulum, guitar, bungee jumping, and the series of motions that have their restoring force opposite the displacement.

Equation of SHM for Displacement

Assume an object of mass ‘m’ having SHM with mean position ‘x0’ and the displacement ‘x’. by taking into consideration the definition of SHM,

Restoring force α – Displacement      (negative sign indicates that restoring force and displacement are opposite to each other) 

According to Newton’s 2nd law of motion,

Force= Mass × Acceleration

Therefore,

Mass × Acceleration α -displacement

⇒m × A α –x

By differentiating the equation,

md2xdt2   α –x

⇒ md2xdt2   = –kx        …… (where k is the force constant)

⇒d2xdt2   = -k × xm

By substituting km as ω2, the equation becomes

d2xdt2   = – ω2 × x

⇒d2xdt2   + ω2 × x = 0….. (Differential equation of SHM)

Multiply differential equation of SHM with 2,

⇒2dxdt  × d2xdt2   + 2dxdt  ω2x= 0

Integrating the above equation,

⇒ (dxdt  )2 + ω2x2 = A (constant)…….(1)

When displacement is to its highest point, 

x = a (where a= amplitude), dxdt  = 0, therefore,

0 + ω2a2 = A 

⇒A= ω2a2

Substituting the value of A in equation (1), 

(dxdt  )2 + ω2x2 = ω2a2

⇒ (dxdt  )2 = ω2a2 – ω2x2

⇒ (dxdt  )2 = ω2 (a2 – x2 )

By taking square root on both sides, we get

Velocity= dxdt  = ω √(a2 – x2 )

Solving for positive,

dxa2  –x2= ωdt

On integrating,

⇒sin-1 xa = ωt + ø(constant)

⇒x =a sin (ωt + ø)

Solving for negative,

dxa2  –x2= -ωdt

On integrating,

⇒cos-1 xa = ωt + Ψ (constant)

⇒ x =a cos (ωt + Ψ)

Both are valid equations if Ψ= ø – 2.

Equation of SHM for velocity

We know that velocity is the ratio of displacement to time.

Therefore, v=dxdt  …….(1)

According to the equation of SHM for displacement,

x =a cos (ωt + Ψ)……(2)

By combining equations 1 and 2, we get

v(t)= ddt  [a cos (ωt + Ψ)]

By doing the derivation, we get

v(t)= -aω sin (ωt + Ψ)

Equation of SHM for displacement

We know that displacement is the ratio of the velocity to time,

Therefore, A=dvdt   …….(1)

According to the equation of SHM for velocity,

v(t)= -aω sin (ωt + Ψ) ……(2)

By combining equations 1 and 2, we get

A = ddt  [ -aω sin (ωt + Ψ)]

By doing the derivation, we get

a(t) = -aω2 cos (ωt + Ψ)

Thus the equations for velocity, displacement, and acceleration in simple harmonic motion are,

x(t)= acos (ωt + Ψ)

v(t)= -aω sin (ωt + Ψ)

A(t) = -aω2 cos (ωt + Ψ)

Terms used in the Simple Harmonic Motion Formula

x = Displacement = Distance between the starting point and endpoint position.

v= velocity= It is the ratio of displacement to time.

A= acceleration= It is the ratio of velocity to time.

a= amplitude= It is the maximum displacement of an object from its fixed position.

ω= angular velocity= Rate of change of angular displacement with time.

t= time= Time taken by an object to complete the oscillation.

Ψ, ø= phase difference = Difference between two different phase angles.

Solved Examples

An SHM along an x-axis Amplitude=.5 m, time to go from one extreme position to other is 2 sec and x=.3m at t=.5 s. Find the general equation of the SHM?

Solution

Let equation of motion is

x=a sin (ωt+ ø)

Now a=0.5m  and Time period= 2×2=4 sec

ω= 2πT = 2

Now it is given, t =0.5sec , x=0.3 m

So,

sin(4+ ø)=350

4+ ø =370

ø =−80

So equation of Motion is,

x= a sin(πt2−80)

Conclusion

Using the equation of SHM, we can conclude the kinetic energy, potential energy, velocity of the object in SHM. The total energy is conserved during the Simple Harmonic Motion.

All the Simple Harmonic Motions are the periodic motion, but all periodic motions are not SHM. By using the equation of velocity, v = ω √(a2 – x2 ) in SHM, when the x=0 that is the object is at the mean position, the velocity will be ωa, i.e. it will be maximum. In contrast, when x=x, that is the object at the extreme position, the velocity will be zero.

Kinetic energy is directly proportional to velocity; thus, it will be maximum at the mean position and zero at the extreme position.

Potential energy is inversely proportional to velocity; thus, it will be maximum at the extreme position and zero at the mean position.