Amplitude of Simple Harmonic Motion (SHM)

Simple harmonic motion is a periodic motion in which a particle move to and fro repeatedly about a mean position in presence of restoring force. Mean position is the central position where particle’s displacement is zero or where particle is at equilibrium position.

 

What is amplitude in SHM?

amplitude & wavelength of shm

Amplitude is the displacement of particle from it’s mean position or equilibrium position. In harmonic motion, amplitude is always directed away from mean position. All periodic motion exhibiting harmonic motion is due to force called restoring force. This restoring force is directly proportional to its amplitude (displacement).
F ∝ -y
or, F = -ky
Where,
F = Restoring force
k = force constant
y = displacement of particles
Negative sign (-ve) represents that restoring force is always directed towards equilibrium position.

Variation of quantities on variation of amplitude

There is variation of quantities like displacement, velocity, acceleration, K.E and P.E of systems exhibiting S.H.M and this variation is due to variation of amplitude at different point of harmonic motion. So, this changes at mean and extreme position can be tabulated as below.


At mean position the value of amplitude is zero but at extreme position its value is equal to the displacement covered (r=y).

Quantity At mean position At Extreme Position
Dsplacement y = 0 (min) y = r (max)
Velocity v = rω v = 0
Acceleration a = 0 a= ω2r
Kinetic Energy K.E = 1/2 mω2r2 K.E = 0
Potential Energy P.E = 0 P.E = 1/2 mω2r2
Total Energy T.E = constant T.E = constant

Amplitude and time period of pendulum

Time period of simple pendulum is given by 
T = 2π√l/g
From above equation,

It is clear that time period of pendulum is independent of amplitude, mass and material of oscillating body.

Time calculation at different amplitude

Let us suppose a particle/body oscillating SHM with an amplitude ‘r’ and time period T. Then time taken to travel different amplitude/displacement can be calculated as related as:

time period of shm at different amplitude

Case I

Time taken to travel A/2 amplitude from mean position

From equation of SHM, 

We have 

y = rsinωt

or, A/2 = rsinωt

or, 1/2 = sinωt

or, ωt = sin-1 (1/2)

or, 2π/T . t = 30°

or, 2π/T . t = π/6

t = T/12

Thus, Time taken to travel A/2 amplitude from mean position is T/12.

 

Case II

Time taken to travel A/2 amplitude from extreme position is T/6.


Case III

Time taken to travel A amplitude from mean position or from extreme position to mean position is T/4 .

Quick Facts

►All particle executing SHM are periodic motion but all periodic motion are not SHM.

►Equation of SHM starting from mean position is y = rsinωt

►Equation of SHM starting from other point but not from mean position is y = rsin(ωt ± φ)

►Equation of simple harmonic motion starting from extreme position is y = rcosωt (φ = 90°).

References

https://en.wikipedia.org/wiki/Simple_harmonic_motion

https://byjus.com/jee/simple-harmonic-motion-shm/

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