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=d^{2}x/dt^{2}
⇒ d^{2}x/dt^{2}=-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
d^{2}x/dt^{2}=-φ^{2}x
(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
cos^{2}θ + sin^{2}θ=1
⇒
A^{2} sin^{2}(ωt+φ)= A^{2}- A^{2}cos^{2}(ωt+φ)
Or
sin(ωt+φ)=[1-x^{2}/A^{2}]
(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=-ω^{2}Acos(ωt+φ) (11)
Equation 11 gives acceleration of particle executing simple harmonic motion and quantity ω^{2} is called acceleration
amplitude and the acceleration of oscillating particle varies betwen the limits ±ω^{2}A.
Putting equation 4 in 11 we get
a=-ω^{2}x
(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(-ω^{2}A or +ω^{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 φ=0.