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SIMPLE HARMONIC MOTION: Definition, Equations, Particle Executing Simple Harmonic Motion

SIMPLE HARMONIC MOTION: DEFINITION, EQUATIONS, RELATIONS EXECUTING SIMPLE HARMONIC MOTION

What-is-Simple-harmonic-motion

What is Simple Harmonic Motion?

Simple Harmonic Motion (SHM) is defined as a to and fro of motion of an object about a mean position. The motion of a body is said to be simple harmonic if its acceleration is always,

a) directed towards a fixed point (mean position)

b) directly proportional to the distance from a fixed point.  

When a body is displaced in simple harmonic motion, a restoring force is set up. Due to this restoring force, the body comes back to its equilibrium, it gets kinetic energy and the body moves away from the equilibrium position. In this way, the body continuously oscillates back and forth about the mean position. Therefore, the motion is said to be a Simple Harmonic Motion (SHM).

Relation between acceleration and displacement of a particle executing simple harmonic motion (SHM):

Consider a particle having mass 'm' moving with a uniform circular motion with an angular velocity 'ω' in a circle of radius 'r' and center at 'O' as shown in the figure. If 'P' be the position of the particle at a time 't'. Draw 'P' perpendicular to OY then 'Q' is a foot of perpendicular drawn from point 'P' on the y-axis.

As the particle moves from point 'P' to 'R', 'Q' moves from point 'O' to point 'R' along the vertical diameter. When the particle moves from 'R' to 'S', 'Q' comes back from point 'R' to point 'O'. As the particle moves from point 'S' to point 'T', 'Q' moves from point 'O' to 'T' and when the particle moves from point 'T' to point 'A', 'Q' comes back from point 'T' to point '0' If the particle continuously goes around the circle, the projection 'Q' goes on vibrating to and fro about the center 'O' of the circle. 

Thus, the foot of perpendicular has to and fro motion along the diameter is simple harmonic motion (SHM).

Now from the shown figure,

Simple-Harmonic-Motion
sinθ = OQ / OP
OQ = OP sinθ 
i.e.  y = r sinθ ⇢ (i)
                where y is the displacement of point Q from the mean position.
y = r sin Ï‰t  [∵ Î¸ = Ï‰t ]
v = rω cos Ï‰t ⇢ (ii)

Further,
Simple-Harmonic-Motion-Equation
Simple-Harmonic-Motion-Equation
Simple-Harmonic-Motion-Equation
Simple-Harmonic-Motion-Equation⇢ (iii)   [∵ using equation (i)]

Now, acceleration is the rate of change in velocity
Simple-Harmonic-Motion-Equation

Equation (iv) is the equation relating the acceleration and displacement of a particle executing simple harmonic motion.
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Tags: simple harmonic motion, simple harmonic motion equation, simple harmonic motion definition, simple harmonic motion formula, a particle executes simple harmonic motion
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