Equation of SHM|Velocity and acceleration|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 it’s 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.
  

Concept Map for SHM

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