Simple Harmonic Motion.

                Before studying Simple Harmonic Motion (SHM) we need to know something called vibration and restoring force.  A particle will vibrate if any displacement from its position of rest calls into a force which tries to bring the particle back to its rest position. This type of force is known as restoring force, which is generally a function of displacement.

                Restoring force F(x) = a₀+a₁x+a₂x²+a₃x³ +..................................

when x=0  i.e  the position of rest, F(x) is also zero so a₀ must be equal to zero. For small displacement, we can restrict the series up to the first order term. Now it becomes F(x)= a₁x. The vibration of this particular type for which restoring force is directly proportional to the displacement is called Simple Harmonic Motion (SHM).

Resorting force and displacement

                For SHM F(x) = -sx (where s is constant and the negative sign is due to the fact that restoring force is directed opposite to the displacement ). A particle vibrating in SHM is known as Linear Harmonic Oscillator (LHO).

                From Newton law of motion we can write F= m x''       here double prime indicates double derivative with respect to time.  so now we have,

                                                                                mx" = -sx

                                                                  hence,   x''= -ω²x  where  ω²=  s/m

                                                                                      x'' +ω²x=0  ............................... (1)

we may write the solution of eq (1) in many ways, one possible solution is  x(t)=acos(ωt +α) ....... (2)  where a is the amplitude and α is the initial phase of the SHM. The period T (time taken for one complete oscillation ) is given by  2π/ω i.e  T = 2π sqrt(m/s).

[ verify yourself  that after time T, x(t) comes back to its initial position i.e x(t+T)=x(t) ]

particle undergoing SHM

Energy of SHM ::

                                velocity of the oscillator v=  -aωsin(ωt +α) .......... (3)

                                acceleration   f = -aω² cos(ωt +α) = -ω²x(t) ................(4)

acceleration is proportional to the displacement and directed opposite to the displacement.

                            Kinetic Energy  E_k = mv² = ma²ω²sin²(ωt +α)  (from eq. (3) )  .................. (5)

                          Potential Energy E_p = mω²a²cos²(ωt +α)  .................... (6)   [ using (1) & (2)]

                                Total energy    E = E_k+E_p = ma²ω²  =constant   .................. (7)

Problems::

1.Show that the projection of a particle moving in a uniform circular motion is a SHM .

Hint::      

                      

Ref..  Vibrations and Waves (The M.I.T. Introductory Physics Series) - A.P.French.  [  use this link.:: click here.]

2.A particle of mass 10g is placed in potential field given by V = (50x²+100) erg/g. what will be the frequency of oscillation?

solution ::  F= -  = -100x = -sx

                        so, s = 100 erg/cm

                                ω²= s/m = 10

                                ω= sqrt(10)  s^-1

Some Simple Examples of SHM ::

1.Simple pendulum:  A simple pendulum is consists of a massless string of length L attached at the top to a rigid support and at the bottom connected to a bob of mass M. Let ψ be the angle made by the string with vertical ( see the figure).

The tangential displacement of the bob = Lψ

The tangential velocity of the bob = Lψ'

The tangential acceleration  of the bob = Lψ''       
 [Here the single and double prime denotes first and second derivative with respect to time respectively ]

Now we can write according to the Newton's law of Motion

                                    Restoring force F =  MLψ''

                                    -Mgsinψ = MLψ'' ........................... (8)

for small values of ψ (in rad ) sinψ = ψ

                        Hence, Lψ'' = - g ψ ................. (9)    

Here also the acceleration is proportional and oppositely directed to the displacement. So according to eq. (4) it is oscillating in SHM.

                                    ψ'' = - (g/L)ψ

                                    ψ'' + (g/L)ψ = 0 ................. (10)

comparing eq.(10) with eq.(1) we can get the frequency of oscillation  ω = sqrt(g/L) hence Time period of oscillation is T = 2π/ ω  = 2π*sqrt(L/g)

                       

                                                                       

For more examples ::
 ref. link-1: Waves : Berkeley Physics Course 1st Edition (Volume 3) -Franks S. Crawford


Summary and Formulas::

1. Restoring Force is always a function of displacement and oppositely directed.
2.  Equation of motion for a particle undergoing SHM is    x'' +ω²x=0.
3. Acceleration of the oscillating particle is always negative and given by  f=-ω²x(t).
4. Angular frequency ω= 2π/T , where T is the time period of oscillation.
5. Total energy of SHM is given by E= ma²ω²  =constant   where a is the amplitude of oscillation and m is the mass of the particle.