Simple Harmonic Motion [Part-II]

The maximum displacement from equilibrium (x = +A or x = -A)

What is the amplitude (A) in SHM?

It oscillates between x = +A and x = -A with period T.

What does the block do after being released from x = +A

x = 0, where the net force equals zero.

What is the equilibrium position in SHM with a spring?

A cosine or sine function with amplitude A and period T

What does a position vs. time graph of SHM resemble?

x(t) = Acos((2π/T)t) = Acos(ωt)

Equation for position as a function of time in SHM

Angular frequency (ω)

ω = (2π/T)

Because the maximum value of cosine is 1, and multiplying by A sets the motion’s maximum displacement.

Why multiply cosine by amplitude A?

The block starts at maximum displacement with zero initial velocity.

What does the initial condition x(0) = 0 and v(0) = 0 represent?

Phase shift (Φ)

A horizontal shift in the cosine/sine function to account for different initial conditions (in SHM)

x(t) = Acos(ωt + Φ)

The generalized equation of SHM with phase shift

Radians per second

Unit for ω

Radians

Unit for Φ

They differ only by a phase shift.

How are sine and cosine functions related in SHM modeling?

v(t) = dx/dt = d/dt(Acos(ωt + Φ)) = -Aωsin(ωt + Φ) = -vmaxsin(ωt + Φ)

The equation for velocity in SHM

At the equilibrium position (x = 0)

Where is velocity maximum in SHM?

vmax = Aω

Equation for maximum velocity in SHM

a(t) = dv/dt = d/dt(-Aωsin(ωt + Φ)) = -A(ω^2)cos(ωt + Φ) = -amaxcos(ωt + Φ)

Equation for acceleration in SHM

amax = Aω^2

Equation for the maximum acceleration in SHM

At maximum displacement (x = +A or x = -A)

Where does maximum acceleration occur?