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Things to remember about Simple Harmonic Motion
- Simple harmonic motion is simplest form of oscillatory motion
- SHM is a kind of motion in which the restoring force is propotional to the displacement from the mean position and opposes its increase.Mathematically restoring force is
F=-Kx
K=Force constant
x=displacement of the system from its mean or equilibrium position
Diffrential Equation of SHM is
d2x/dt2 + ω2x=0
S - olutions of this equation can both be sine or cosine functions .We conveniently choose
x=Acos(ωt+φ)
where A,ω and φ all are constants
- Quantity A is known as amplitude of SHM which is the magnitude of maximum value of displacement on either sides from the equilibrium position
- Time period (T) of SHM the time during which oscillation repeats itself i.e, repeats its one cycle of motion and it is given by
T=2π/ω
where ω is the angular frequency
- Frequency of the SHM is the number of the complete oscillation per unit time i.e, frequency is reciprocal of the time period
f=1/T
Thus angular frequncy
ω=2πf
- Total energy remains constant in a SHM.So you can find the energy at any position and differentiate it to find the out the frequency
- Problem of SHM are basically to find out the timeperiod.So the concenteration should be on getting the net restoring force
- The basic approach to solve such problem is
1. Consider the system is displaced from equilibrium position
2. Now consider all the forces acting on the system in displaced position
3. find the restore force which comes out to be in the form
4.F=-kx
Things to remember about Waves
- Transverse waves are such waves where the displacements or oscillations are perpandicular to the direction of propagation of wave.
- Longitudinal waves are those waves in which displacement or oscillations in medium are parallel to the direction of propagation of wave for example sound waves.
- At any time t , displacement y of the particle from it's equilibrium position as a function of the coordinate x of the particle is
y(x,t)=A sin(ωt-kx)
where,
A is the amplitude of the wave
k is the wave number
ω is angular frequency of the wave
and (ωt-kx) is the phase.
- Wavelength λ and wave number k are related by the relation
k=2π/λ
- Time period T and frequency f of the wave are related to ω by
ω/2π = f = 1/T
- speed of the wave is given by
v = ω/k = λ/T = λf
- Principle of superposition:
When two or more waves traverse thrugh the same medium,the displacement of any particle of the medium is the sum of the displacement that the individual waves would give it.
y=Σyi(x,t)
- The interference of two identical waves moving in opposite directions produces standing waves.
- For a string with fixed ends standing wave is given by
y(x,t)=[2Acos(kx)]sin(ωt)
above equation does not represent travelling wave since it does not have characterstic form involving (ωt-kx) or (ωt+kx) in the argument of trignometric function.
- In standing waves amplitude of waves is different at different points i.e., at nodes amplitude is zero and at antinodes amplitude is maximumwhich is equal to sum of amplitudes of constituting waves.
- At intermediate points amplitude of wave varies between these two limits of maxima and minima
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