If a moving body repeats its motion after regular intervals of time the motion is said to be harmonic or periodic. The time interval after which it repeats the motion is called time period. if the body moves to and fro on the same path, it is said to be oscillation. In simple harmonic motion the particle moves in a straight line and the acceleration of the particle is always infected towards a fixed point on the line. This fixed point is called centre of oscillation. The acceleration in SHM is given by
a = - ω2x or F = - mω2x = - Kx
K = mω2 is called the force constant or spring constant.
The force which brings the particle back towards the equilibrium (or mean) position is called restoring force.
SHM may be thought of as the projection of uniform circular motion along a diameter.
X = r cos αx
Y = r sin αx
Α = ω2 x
Or d2x / dt2 = - ω2x
Given the solution x = x0sin αx
When the particle starts from mean position,
X = x0 cos ωt
When, starts from extreme position
X = x0 sin (ωt ± Ø)
X = x0 cos (ωt ± Ø)
When it starts in between mean and extreme positions.
X = x0 e ±iωt
Where x is called instantaneous displacement, x0 the amplitude (maximum displacement) Ø initial phase angle or epoch or angle of repose and ω the angular frequency.
Linear frequency f = ω / 2π and T = 1 / ƒ is time period
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