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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
Â Â Â Â Â Â Â Â Â Â a=F/m
Â Â Â Â Â Â Â Â Â Â =-kx/m
but we know that acceleration a=dv/dt=d^{2}x/dt^{2}
â‡’ Â Â Â Â Â Â Â Â Â Â d^{2}x/dt^{2}=-kx/m
Â Â Â Â Â Â Â Â Â Â (1)
This equation 1 is the equation of motion of SHM.
- If we choose a constant Ï†=âˆš(k/m) then equation 1 would become
Â Â Â Â Â Â Â Â Â Â d^{2}x/dt^{2}=-Ï†^{2}x
Â Â Â Â Â Â Â Â Â Â (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
Â Â Â Â Â Â Â Â Â Â x=A sinÏ‰t
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (3a)
Â Â Â Â Â Â Â Â Â Â x=A cosÏ‰t
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (3b)
Â Â Â Â Â Â Â Â Â Â x=A cos(Ï‰t+Ï†)
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (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,
Â Â Â Â Â Â Â Â Â Â x=A cos(Ï‰t+Ï†)
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (4)
and A , Ï† and Ï† are all constants.
- Fig below shows the displacement vs. time graph for phase Ï†=0.
4. Characterstics of SHM
Here in this section we will learn about physical meaning of quantities like A, T, Ï‰ and Ï†.
(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,
Â Â Â Â Â Â Â Â Â Â cos(Ï‰t+Ï†)=cos(Ï‰(t+T)+Ï†)
cosine function cos(Ï‰t+Ï†) will repeat it's value if angle (Ï‰t+Ï†) is increased by 2Ï€ or any of it's multiple. As T is the pime period
Â Â Â Â Â Â Â Â Â Â (Ï‰(t+T)+Ï†)=(Ï‰t+Ï†)+2Ï€
or,Â Â Â Â Â Â Â Â Â Â T=2Ï€/` = 2Ï€âˆš(m/k)
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (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.
Â Â Â Â Â Â Â Â Â Â f=1/T = 1/2Ï€(âˆš(k/m))
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (6)
Thus, Â Â Â Â Â Â Â Â Â Â Ï‰=2`/T = 2`f
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (7)
- This quantity Ï‰ is called the angular frequency of SHM.
- S.I. unit of T is s (seconds)
Â Â Â Â Â f is Hz (hertz)
Â Â Â Â Â Ï‰ is rad s^{-1} (radian per second)
(c) Phase
- Quantity (Ï‰t+Ï†) in equation (4) is known as phase of the motion and the constant Ï† 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 Ï†=0 then from equation 4
Â Â Â Â Â Â Â Â Â Â x=A cosÏ‰t
that is displacement of oscillating particleis maximum , equal to A at t=0 when the motion was started. Again if Ï†=`/2 then from equation 4
Â Â Â Â Â Â Â Â Â Â x=A cos(Ï‰t+`/2)
Â Â Â Â Â Â Â Â Â Â =AsinÏ‰t
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
Â Â Â Â Â v=dx/dt
- In SHM displacement of particle is given by
Â Â Â Â Â x=A cos(Ï‰t+Ï†)
now differentiating it with respect to t
Â Â Â Â Â v=dx/dt= AÏ‰(-sin(Ï‰t+Ï†))
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (8)
- Here in equation 8 quantity AÏ‰ is known as velocity amplitude and velocity of oscillating particle varies between the limits Â±Ï‰.
- From trignometry we know that
Â Â Â Â Â cos^{2}Î¸ + sin^{2}Î¸=1
â‡’
Â Â Â Â Â A^{2} sin^{2}(Ï‰t+Ï†)= A^{2}- A^{2}cos^{2}(Ï‰t+Ï†)
Or
Â Â Â Â Â sin(Ï‰t+Ï†)=[1-x^{2}/A^{2}]
Â Â Â Â Â Â Â Â Â Â (9)
putting this in equation 8 we get,
- From this equation 10 we notice that when the displacement is maximum i.e. Â±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 Ï†=0.
6. Acceleration of SHM
- Again we know that acceleration of a particle is given by
Â Â Â Â Â a=dv/dt
where v is the velocity of particle executing motion.
- In SHM velocity of particle is give by,
Â Â Â Â Â v= -Ï‰sin(Ï‰t+Ï†)
differentiating this we get,
or,
Â Â Â Â Â a=-Ï‰^{2}Acos(Ï‰t+Ï†)Â Â Â Â Â Â Â Â Â Â (11)
- Equation 11 gives acceleration of particle executing simple harmonic motion and quantity Ï‰^{2} is called acceleration
amplitude and the acceleration of oscillating particle varies betwen the limits Â±Ï‰^{2}A.
- Putting equation 4 in 11 we get
Â Â Â Â Â a=-Ï‰^{2}xÂ Â Â Â Â Â Â Â
Â Â Â Â Â Â Â Â Â Â Â Â (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(-Ï‰^{2}A or +Ï‰^{2}A)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 Ï†=0.
Concept Map for SHM