Oscillations Flashcards
What is periodic motion?
Any motion that repeats itself over and over again at regular intervals of time
Oscillatory Motion / Harmonic
Body moves back and forth repeatedly about its mean position
Fourier theorem
Any arbitrary function F(t) with period T can be expressed as the unique combination of sine and cosine functions fn (t) and gn (t) with suitable coefficients
Harmonic Functions
Periodic functions that can be represented by a sine or cosine curve
Non-harmonic Functions
Periodic functions which cannot be represented by single sine or cosine function
Simple Harmonic Motion
If it moves to and fro about a mean position under the action of a restoring force which is directly proportional to its displacement from the mean position and is always directed towards the mean position
Prove to find the differential equation of SHM
-
d^x / dt^2 + (omega)^2 x = 0
Find the of SHM:
1. Displacement
2. Time period
- x = A cos(omega t + psi not)
- T = 2pi (root m / k)
When to use x = A sin omega t and x = A cos omega t
x = A cos omega t –> When it starts from maximum displacement
x = A sin omega t –> When it starts from equilibrium position
Value of Spring Constant
k = m omega^ 2 = F / x
Harmonic Oscillator
A particle executing simple harmonic motion
Amplitude
Maximum displacement of the osciallating particle on either side of its mean position
Oscillation
One complete back and forth motion of a particle starting and ending at the same point
Angular frequency
Quantity obtained by multiplying frequency by a factor of 2pi
Phase
Gives the state of the particle as regards its position and the direction of motion at that instant
Initial phase / Epoch
Phase of a vibrating particle corresponding to time t = 0
Circle of Reference
A circle of reference can refer to a circle used to establish the center of a component or to a circle used to describe the motion of a particle undergoing simple harmonic motion (SHM)
Displacement formula from centre of circle
x = A cos (omega t + psi not)
A = amplitude
x = Displacement
omega t = angular frequency
psi not = initial phase
Acceleration in SHM
a(t) = - omega ^2 A cos (omega t + psi not) = - omega ^2 x
Acceleration amplitude
Acceleration amplitude is the maximum acceleration of an oscillating particle
Acceleration amplitude formula
a max = omega^2 A = (2pi / T)^2 A
acceleration in SHM
a(t) = omega ^2 A cos (omega t + pi)
Kinetic Energy in SHM
KE = 1/2 m omega^2 A^2 sin^2 (omega t + psi not) = 1/2 m omega^2 (A^2 - x^2)
Potential Energy in SHM
U = 1/2 kx^2 = 1/2 m omega^2 x^2 = 1/2 m omega^2 A^2 cos^2 (omega t + psi not)
Total Energy in SHM
E = 1/2 k A^2 = 1/2 m omega^2 A^2 = 2 pi^2 m v^2 A^2
Time period in SHM
T = 2pi root (m / k)
Oscillation frequency for springs connected in parallel
mu = 1/2pi [root (k1 + k2) / m]
Oscillation frequency for springs connected in series
mu = 1 / 2pi [root (k1k2 / m (k1 + k2))]
Simple Pendulum
Consists of a point-mass suspended by a flexible, inelastic and weightless string from a rigid support of infinite mass
Time period of a bob of simple pendulum
T = 2pi root (l / g)
Free Oscillations
If a body, capable of oscillation, is slightly displaced from its position of equiliibrium and left to itself, it starts oscillating with a frequnecy of its own
Damped Oscillations
Oscillations in which the amplitude decreases gradually with the passage of time
