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Simple Harmonic Motion (SHM)


3. Equation of SHM

  • Consider any particle executing SHM with origin as it's equilibrium position under the influence of restoring force F=
  • kx , where k is the force constant and x is the displacement of particle from the equilibrium position.
  • Now since F= -kx is the restoring force and from Newton's law of motion force is give as F=ma , where m is the mass of the particle moving with acceleration a. Thus acceleration of the particle is
              a=F/m
               =-kx/m

    but we know that acceleration a=dv/dt=d2x/dt2
    ⇒           d2x/dt2=-kx/m           (1)
    This equation 1 is the equation of motion of SHM.
  • If we choose a constant φ=√(k/m) then equation 1 would become
              d2x/dt2=-φ2x           (2)
  • This equation is a differential equation which says that displacement x must be a funcyion of time such that when it's second derivative is calculated the result must be negative constant multiplied by the original function.
  • Sine and cosine functions are the functions satisfying above requirement and are listed as follows
              x=A sinωt                          (3a)
              x=A cosωt                          (3b)
              x=A cos(ωt+φ)                     (3c)
    each one of equation 3a, 3b and 3c can be submitted on the left hand side of equation 2 and can then be solved for varification.
  • Convinently we choose equation 3c i.e., cosine form for representing displacement of particle at any time t from equilibrium position. Thus,
              x=A cos(ωt+φ)                     (4)
    and A , φ and φ are all constants.
  • Fig below shows the displacement vs. time graph for phase φ=0.

4. Characterstics of SHM

Here in this section we will learn about physical meaning of quantities like A, T, ω and φ.
(a) Amplitude
  • Quantity A is known as amplitude of motion. it is a positive quantity and it's value depends on how oscillations were started.
  • Amplitude is the magnitude of maximum value of displacement on either side from the equilibrium position.
  • Since maximum and minimum values of any sine and cosine function are +1 and -1 , the maximum and minimum values of x in equation 4 are +A and -A respectively.
  • Finally A is called the amplitude of SHM.
(b) Time period
  • Time interval during which the oscillation repeats itself is known as time period of oscillations and is denoted by T.
  • Since a particle in SHM repeats it's motion in a regular interval T known as time period of oscillation so displacement x of particle should have same value at time t and t+T. Thus,
              cos(ωt+φ)=cos(ω(t+T)+φ)
    cosine function cos(ωt+φ) will repeat it's value if angle (ωt+φ) is increased by 2π or any of it's multiple. As T is the pime period
              (ω(t+T)+φ)=(ωt+φ)+2π
    or,          T=2π/` = 2π√(m/k)                (5)
  • Equation 5 gives the time period of oscillations.
  • Now the frequency of SHM is defined as the numberof complete oscillations per unit time i.e., frequency is reciprocal of time period.
               f=1/T = 1/2π(√(k/m))                 (6)
    Thus,            ω=2`/T = 2`f                      (7)
  • This quantity ω is called the angular frequency of SHM.
  • S.I. unit of T is s (seconds)
          f is Hz (hertz)
          ω is rad s-1 (radian per second)
(c) Phase
  • Quantity (ωt+φ) in equation (4) is known as phase of the motion and the constant φ is known as initial phase i.e., phase at time t=0, or phase constant.
  • Value of phase constant depends on displacement and velocity of particle at time t=0.
  • The knowledge of phase constant enables us to know how far the particle is from equilibrium at time t=0. For example,
    If φ=0 then from equation 4
              x=A cosωt
    that is displacement of oscillating particleis maximum , equal to A at t=0 when the motion was started. Again if φ=`/2 then from equation 4
              x=A cos(ωt+`/2)
               =Asinωt

    which means that displacement is zero at t=0.
  • Variation of displacement of particle executing SHM is shown below in the fig.


5.Velocity of SHM

  • We know that velocity of a particle is given by
         v=dx/dt
  • In SHM displacement of particle is given by
         x=A cos(ωt+φ)
    now differentiating it with respect to t
         v=dx/dt= Aω(-sin(ωt+φ))                (8)
  • Here in equation 8 quantity Aω is known as velocity amplitude and velocity of oscillating particle varies between the limits ±ω.
  • From trignometry we know that
         cos2θ + sin2θ=1

         A2 sin2(ωt+φ)= A2- A2cos2(ωt+φ)
    Or
         sin(ωt+φ)=[1-x2/A2]            (9)
    putting this in equation 8 we get,


  • From this equation 10 we notice that when the displacement is maximum i.e. ±A the velocity v=0, because now the oscillator has to return to change itís direction.
  • Figure below shows the variation of velocity with time in SHM with initial phase φ=0.

6. Acceleration of SHM

  • Again we know that acceleration of a particle is given by
         a=dv/dt
    where v is the velocity of particle executing motion.
  • In SHM velocity of particle is give by,
         v= -ωsin(ωt+φ)
    differentiating this we get,

    or,
         a=-ω2Acos(ωt+φ)          (11)
  • Equation 11 gives acceleration of particle executing simple harmonic motion and quantity ω2 is called acceleration amplitude and the acceleration of oscillating particle varies betwen the limits ±ω2A.
  • Putting equation 4 in 11 we get
         a=-ω2x                     (12)
    which shows that acceleration is proportional to the displacement but in opposite direction.
  • Thus from above equation we can see that when x is maximum (+A or -A), the acceleration is also maximum(-ω2A or +ω2A)but is directed in direction opposite to that of displacement.
  • Figure below shows the variation of acceleration of particle in SHM with time having initial phase φ=0.


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