Periodic motion
What are the terms used in oscillation:
a = Fx/m
a = acceleration
Fx=force exerted on a body in x direction
m=mass
relationship between frequency and period:
frequency (f) = number of cycles in one second
expressied as 1/T
T is period
therefore T = 1/f
angular frequency and velocity:
f = 1/T T=1/f
ω = 2πf = 2π/T
V = fλ (unit: m/s)
simple harmonic motion:
F=−kx, with k being the spring constant and x the displacement from the equilibrium position.
Equation for SHM - Displacement
x = Acos𝜽
x = Acosωt
Equation for SHM – velocity
𝒗𝒙 = −𝒗𝑸 𝒔𝒊𝒏𝝎𝒕 = −𝝎𝑨𝒔𝒊𝒏𝝎𝒕
Equation for SHM - Acceleration
𝑎𝑥 = −𝑎𝑄 cos𝜃
𝑎𝑥 = −𝜔2𝐴
Angular frequency (ω) for SHM
a = -(k/m)x
a = -ω²x
ω2 = k/m
ω =( k/m )^(1/2)
Displacement, Velocity and Acceleration with phase angle:
𝒙 = 𝑨 cos(𝝎𝒕 + 𝝋)
A = amplitude
t = angular fequency
𝝋 = phase angle
Phase angle values and x
𝒙𝒐 = 𝑨 cos𝝋
To find amplitude and phase angle
𝒗𝒐 = − 𝝎𝑨 sin𝝋
Energy in SHM
𝑬 =𝟏/𝟐 𝒎𝒗𝟐 +𝟏/𝟐 𝒌𝒙^𝟐 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝑬 =𝟏/𝟐 𝒎𝒗𝟐 +𝟏/𝟐 𝒌𝒙𝟐 =𝟏/𝟐 𝒌𝑨^𝟐 =𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝒗 = ±(𝒌/𝒎)^(𝟏/𝟐) (𝑨^𝟐−𝒙^𝟐)^(𝟏/𝟐)