- Periodic Motion
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- Simple Harmonic Motion
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- Equation of SHM
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- Characterstics of SHM
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- Velocity of SHM
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- Acceleration of SHM
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- Total Energy of SHM
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- Motion of a body suspended from a spring
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- Simple pendulum
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- The compound pendulum
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- Damped Oscillations
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- Driven or Forced Harmonic oscillator

- 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.

- 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.

- 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)

- 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.

- 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 its direction.
- Figure below shows the variation of velocity with time in SHM with initial phase φ=0.

- 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.

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