SIMPLE HARMONIC MOTION

SIMPLE HARMONIC MOTION

1. Essential conditions of SHM

   • 1. f=-x , where f is linear acceleration and x is displacement
     • 2. α = -θ , where α is the angular acceleration and θ is angular displacement

2. If the acceleration of a particle motion is always directly proportional to its displacement from it equilibrium (or mean) position & is directed towards equilibrium position, then it is called as SHM.

   • Differential equation of SHM: -Consider a particle of mass ‘m’ executing an SHM along a straight line, x is the displacement from the mean position at ‘t’. -The restoring force F will be proportional to the displacement & will be directed opposite to this. F ∝ -x F= -kx ----- (1) Where k is the proportionality constant/force constant If f= d2x/dt2 be the acceleration at any instant of time, then m d2x/dt2 = -kx ------(2) m d2x/dt2 + kx = 0 d2x/dt2 + k/m = 0 Now, put k/m = ω2x d2x/dt2 + ω2x = 0 ------- (3) Eqn (2) & (3) are the differential equation for l simple harmonic motion.
     • The desired solution from the differential equation of SHM, which gives the displacement of particle executing SHM at any instant of time. X = A sin(ωt+ δ)
       • Time Period: The shortest time taken to complete one oscillation is also defined as the time period. T = 2π/ω
       • Frequency: The number of oscillations per second is defined as the Frequency. We know angular frequency ω = 2πf = 2π/T Therefore, n = 1/T = ω/2π = 1/2π√(k/m)
       • Phase in SHM: The phase of a vibrating particle at any instant is the state of the vibrating (or) oscillating particle regarding its displacement and direction of vibration at that particular instant. The expression, position of a particle as a function of time. x = A sin (ωt + Φ) Where (ωt + Φ) is the phase of the particle, the phase angle at time t = 0 is known as the initial phase.
       • Acceleration of SHM: The expression for acceleration of the oscillating particle at any displacement which is the standard equation of SHM is given by a = d2x/dt2 = – ω2 x o r f = – ω2 x
         • - At extreme position, x=A therefore, fmax= ω2 A - At mean position, x=0 therefore, fmin=0
       • Velocity of SHM: The expression for velocity of the particle at any displacement x is given as, v = ± ω √a2 – x2
         • - At mean position i.e. x=0 the particle has maximum velocity, given as vmax = ωA - At extreme position i.e. displacement x is maximum and velocity is zero, given as x=A

3. DAMPED HARMONIC OSCILLATIONS

   • Free Vibrations: The body keeps on vibrating indefinitely & such vibrations are called as Free Vibrations. But, there is always some resistance offered to the oscillating system.
   • Damped Vibrations: If a body set into vibrations will have its amplitude continuously decreasing due to frictional resistance & then the vibrations will die after some time. Then the motion is said to be damped by the friction & so it is called as damped vibrations.
   • Suppose, a system has a body of mass ‘m’ attached to a spring whose force constant k. Consider x’ is the displacement of the body. Then dx’/dt is the instantaneous velocity. Therefore, two types of forces acting on the body: i) Restoring Force (Fr = -kx’) is proportional to x’ and act in opposite direction to the displacement. and ii) A damping force is –q’ dx’/dt is proportional to the velocity and is oppositely directed to the motion.
     • Then the total force acting on Harmonic Oscillator is: d2x’/dt2 + 2s dx’/dt + ω2x’ = 0
     • And the total energy acting on Harmonic Oscillator is: E = ½ m ω2A2

4. Linear SHM: When a particle moves to and fro about a fixed point (called equilibrium position) along with a straight line then its motion is called linear Simple Harmonic Motion. For Example: spring-mass system

5. Angular SHM: When a system oscillates angular long with respect to a fixed axis then its motion is called angular simple harmonic motion.

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