S.H.M As A Projection of Uniform Circular Motion | STD 11 Physics

Simple Harmonic Motion (S.H.M.) is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the direction opposite to that of displacement.
For an object undergoing S.H.M., several equations can be derived:
Displacement (x) as a function of time (t):
x(t)=Asin⁡(ωt+ϕ)
Where:
x(t) is the displacement at time t.
A is the amplitude of the motion.
ω is the angular frequency given by ω=2πf, where ff is the frequency of oscillation.
ϕ is the phase constant.
Velocity (v) as a function of time (t):
v(t)=dx/dt=Aωcos⁡(ωt+ϕ)
This equation gives the velocity of the object at any time t.
Acceleration (a) as a function of time (t):
a(t)=dv/dt=−Aω2sin⁡(ωt+ϕ)
This equation provides the acceleration of the object at any time t. Note the negative sign indicating that the acceleration is directed opposite to the displacement.
Force (F) as a function of displacement (x):
F=−kx
Where:
k is the spring constant or stiffness constant of the spring or system.
x is the displacement from the equilibrium position.
The negative sign in the force equation signifies that the force acts in the direction opposite to the displacement, which is characteristic of restoring forces in S.H.M.
These equations form the foundational mathematical description of simple harmonic motion and are derived from Newton's second law of motion (F = ma) and Hooke's law for springs (F = -kx).
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