Uniform Circular Motion & SHM | Derivation of X, V, & a | SIMPLE HARMONIC MOTION | Physics 12

Uniform Circular Motion & SHM | Derivation of X, V, & a | SIMPLE HARMONIC MOTION | Physics 12 | National Book Foundation | Federal board
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In this comprehensive lecture from Chapter 17: Simple Harmonic Motion, we explore the deep connection between Uniform Circular Motion and Simple Harmonic Motion (SHM) as explained in the National Book Foundation textbook for Class 12 (Federal Board).

The motion of a body in SHM can be understood by analyzing the projection of a point moving in uniform circular motion onto a diameter of the circular path. If a point P moves in a circle of radius x₀ at constant angular velocity ω, then the projection Q of point P on the x-axis performs SHM between –x₀ and +x₀. The time period of this oscillation is equal to the time for one complete revolution of P around the circle.
Displacement Derivation:
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At any instant t, the angle θ made by OP with the x-axis is:
θ = ωt
The displacement x of point Q is the horizontal component:
x = x₀ cos(ωt) (Eq. 17.5)
This is the standard equation of SHM and shows that Q performs simple harmonic motion.

Velocity Derivation:
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The velocity of point P is tangential and has magnitude:
vₚ = constant
The velocity of Q is the horizontal component of vₚ:
v = vₚ sin(θ)
Substitute θ = ωt and vₚ = ωx₀:
v = ωx₀ sin(ωt)
Using the identity: sin²θ = 1 – cos²θ and x = x₀ cos(ωt)
We get:
v = ω √(x₀² – x²)
This gives the instantaneous velocity of SHM at displacement x.
Acceleration Derivation:
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The centripetal acceleration of point P is:
aₚ = x₀ ω²
The horizontal component (acceleration of Q) is:
a = –aₚ cos(θ)
= –x₀ ω² cos(ωt)
Using cos(ωt) = x / x₀, we get:
a = –ω² x
This shows that acceleration is proportional to –x and directed towards the mean position, satisfying the SHM condition.
Conclusion:
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Thus, when a body moves in a circle, its projection on a diameter undergoes simple harmonic motion. The expressions for displacement, velocity, and acceleration of SHM are:
Displacement: x = x₀ cos(ωt)
Velocity: v = ω √(x₀² – x²)
Acceleration: a = –ω² x
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