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Mechanical Waves, Simple Harmonic Motion motion that repeats itself in a…
Mechanical Waves
Types of Mechanical Waves
Transverse
: Displacement is perpendicular
to the direction of motion
Longitudinal
: Displacement is equal direction of motion
Periodic Transverse Wave
ω
=2πf
T
= 2π/f
Speed Of Periodic Wave
v
= λf
Periodic Longitudinal Wave
Mathematical Description Of A Wave
y
(
x
,
t
) =
Acosωt
=
Acos2πft
when
x
≠
0
, but moves to the right with time
t
=
x/v
, motion of
x
at time
t
can be described as motion at
x
=
0
at an earlier time of
t
-
x/v
y(x,t) = A cos[ω(t-x/v)]
with
cos
(-θ) =
cos
θ, y(x,t) = A cos[ω(x/v
Simple Harmonic Motion
motion that repeats itself in a regular cycle
Hooke's Law
F=-kx
Key Characteristics
Amplitude (A): Maximum displacement from equilibrium.
Period (T): Time for one complete cycle.
Frequency (f): Number of cycles per unit time (f = 1/T).
Phase: Describes the position in the cycle.
Angular Frequency (ω): Related to frequency by ω = 2πf.
Equations of SHM
Displacement: x(t)=Acos(ωt+ϕ)
Velocity: v(t)=−Aωsin(ωt+ϕ)
Acceleration: a(t)= -Aω^2 cos(ωt+ϕ)
Applications of SHM
Mass-spring system
Pendulum (small angle approximation)
Vibrations in musical instruments
Radio waves (resonance)
SHM in Cicular Motion
Energy In SHM
Potential Energy
U = 1/2 kx²
Kinetic Energy
K = 1/2mv²
Total Mechanical Energy
E = U+K = 1/2kx² + 1/2mv²
Pendulum
Two forces acting : string tension T and gravity
Damped Oscillation
An oscillation the runs down and stops-friction-mechanical energy transforms into thermal energy
Resonance
Natural Frequency
Driving Frequency
