Graphical representation of displacement, velocity and acceleration of a particle vibrating simple harmonically with respect to time t.
(i) Displacement graph is a sine curve. Maximum displacement of the particle is y =+a
(ii) The velocity of the vibrating particle is maximum at the mean position i.e v =+ aw and it is zero at the extreme position.
(iii) The acceleration of the vibrating particle is zero at the mean position and maximum at the extreme position (i.e) ± aw².
The velocity is ahead of displacement by a phase angle of \(\frac { \pi }{ 2 } \) . The acceleration is ahead of the velocity by a phase angle \(\frac { \pi }{ 2 } \) or by aw² phase  \(\pi\) ahead of displacement. (i.e) When the displacement has its greatest positive value, acceleration has its negative maximum value or vice versa.

Acceleration in SHM:
The rate of change of velocity is the acceleration of the vibrating particle.
\(\frac { d^{ 2 }y }{ dt^{ 2 } } =\frac { d }{ dt } \left( \frac { dy }{ dt } \right) =\frac { d }{ dt } \) (aw(cos wt))
= w²asin wt
Acceleration =\(\frac { { d }^{ 2 }y }{ dt^{ 2 } } \)= -w²y
The acceleration of the particle can also be obtained by component method. The centripetal acceleration of the particle pacting along po is \(\frac { { v }^{ 2 } }{ a } \)

Acceleration in SHM:
This acceleration is resolved into two components
\(\frac { { v }^{ 2 } }{ a } \) cosθ along PN perpendicular to oy.
\(\frac { { v }^{ 2 } }{ a } \) sinθ along perpendicular to oy.
Hence acceleration = \(\frac { { v }^{ 2 } }{ a } \) sinθ
= -aw²sinwt
= -w²y
Acceleration = -w²y
The negative sign indicates that the acceleration is always opposite to the direction of displacement and is directed toward the centre.
Special cases:
When particle is at the mean position (i.e), y = 0, the acceleration is zero.
When the particle is at the extreme position (i.e) y = ± a acceleration is ± a W² which is called as acceleration amplitude.

Displacement in SHM:
The distance travelled by the vibrating particle at any instant of time t from its mean position is known as displacement. When the particle is at p, the displacement of the particle along y axis is y.

The in ΔOPN, sinθ=\(\frac { ON }{ op } \)
ON = y = op sinθ
y = op sin wt
Since op = a, the radius of the circle, the displacement of the vibrating particle is
y = a sin wt    ....(1)
The amplitude of the vibrating particle is defined as its maximum displacement from the mean position.
Velocity in SHM:
The rate of change of displacement is the velocity of the vibrating particle.
Differentiating equation (1), with respect to time t.
\(\frac { dy }{ dt } =\frac { d }{ dt } \) (a sin wt)    ....(2)
∴ v = aw cost wt
The velocity v of the particle moving along the circle can also be obtained by resolving it into two components.

Velocity in SHM
(i) cos θ in a direction parallel to oy.
(ii) V sin θ in a direction perpendicular to oy.
The component v sin θ has no effect along yoy' since it is perpendicular to oy
Velocity =V cosθ  = V cos wt
We know that,
Linear Velocity = radius\(\times\)angular velocity
∴ v = aw
Velocity = aw cost wt
Velocity = \(aw\sqrt { 1-sin^{ 2 }wt } \)
∴ Velocity =\(w=\sqrt { ({ a }^{ 2 }-y^{ 2 }) } \)
Special cases:
When the particle is at mean position,
(i.e.) y = 0. Velocity is aw and is maximum.
v = ± aw is called velocity amplitude.
When the particle is in the extreme position,
(i.e) y = +a, the velocity is zero.

Consider a particle moving along the y circumference of a circle of radius a and NP centre 0, with uniform speed v, in anticlockwise direction.

Let xx¹ and yy¹ be the two perpendicular x diameters. Suppose the particle is at p after a time t. If w is the angular velocity then the angular displacement θ in time t is given by θ = -wt.
From p draw pN perpendicular to yy¹. As the particles moves from x to y, foot of the perpendicular N moves from 0 to y. As it moves further from y to x¹, then from x¹ to y¹ and back again to x, the point N moves from y to 0, from 0 to y¹ and back again to O. When the particle completers one revolution along the circumference, the point N completes one vibration about the mean position O. The motion of the point N along the diameter yy¹ is simple harmonic.
Hence the projection of a uniform circular motion on a diameter of side in simple harmonic motion.

We shall first discuss the general dynamics of an SHM. We know that work has to be done on a system to displace it from its position of equilibrium. The restoring force F obviously depends on the work done to give a displacement x.



(i) Thus, F is some general function of x. For system oscillating violently (large x) the dependence of F on x is very complex.
(ii) This is called small oscillation approximation. For such system, the restoring force is proportional to the displacement and opposes its increase.
F = -kx    ...(1)
(iii) The negative sign indicates that F opposes increase in x. k, the, constant of proportionality, is called the force constant. The Mks unit for k is \(\frac{N}{m}\) . The magnitude of k depends on the elastic properties of the system. In the specific examples, in the system consisting of a mass and a spring, k will depend on the stiffness or strength of the spring.
(iv) The equation (1) is a statement of Hooke's law for elastic forces. The general definition of SHM is the motion in which the restoring force is proportional to the displacement from the mean position and opposes its increase.
(v) That in such a motion the displacement varies harmonically with time.