Oscillation
- repetitive variations of some measure about a point of equilibrium or between two or more different states
- types: (i) Damped Free Oscillation
- Free oscillation subjected to dissipation forces
- Object will come to rest
- Real life scenario
(ii) Undamped Free Oscillation
- No mechanical energy lost to surroundings
- An oscillator displaced from equilibrium will be subjected to a restoring force
Simple Harmonic Motion
- ideal type of periodic motion in which the body oscillates about an equilibrium position in a sinusidal pattern
- frequency, period and amplitude constant
- Defined as a periodic motion in which the acceleration of the oscillator is directly proportional to the displacement from its equilibrium point and is always directed to that point
- a = - ω^2 x
SHM relationships Displacement - time
(i) Start at x = 0 , with +v
- X = Xo sin ωt
(ii) Start at +Xo , with v = 0
- X = Xo cos ωt
(iii) Start at + X ' , with +v
- X = Xo sin ( ωt + Ф1 )
or
- X =Xo cos ( ωt + Ф2 )
Velocity - time (i) X = Xo sin ωt
- V = ωXo cos ωt
- a = - ω^2Xo sin ωt
(ii) X = Xo cos ωt
- V = ωXo sin ωt
- a = - ω^2Xo cos ωt
Uniform Circular Motion
- Frequency of SHM = frequency of Circular Motion
- Amplitude = radisu of circle
Energy
- KE = 1/2 m ω^2 ( (Xo)^2 - X^2)
- Max KE = 1/2 m ω^2 ( Xo )^2
- PE = TE - KE
- PE = 1/2 m ω^2 x^2
-Max PE = 1/2m ω^2 (Xo)^2
Damped Oscillations - Progressive decrease in amplitude
- Examples:
(i) Air resistance
- increase surface area
(ii) Fluid Viscosity
(iii) Eddy Current
- Extent of Damping
(i) Light
- Decrease with same proportion
- Stops only after a large number of osciallations
(ii) Critical
- Returns to equilibrium in minimum time
(iii) Heavy
- Returns to equilibrium after a long time
Resonance
- It is a phenomenon in which an oscillatory system reacts with maximum amplitude to an external periodic force whose driving frequency matches the natural frequency of the driven system
- As system gets more heavily damped,
(i) Amplitude decreases
(ii) Resonance peak broader
(iii) Resonance peak shifts left
- repetitive variations of some measure about a point of equilibrium or between two or more different states
- types: (i) Damped Free Oscillation
- Free oscillation subjected to dissipation forces
- Object will come to rest
- Real life scenario
(ii) Undamped Free Oscillation
- No mechanical energy lost to surroundings
- An oscillator displaced from equilibrium will be subjected to a restoring force
Simple Harmonic Motion
- ideal type of periodic motion in which the body oscillates about an equilibrium position in a sinusidal pattern
- frequency, period and amplitude constant
- Defined as a periodic motion in which the acceleration of the oscillator is directly proportional to the displacement from its equilibrium point and is always directed to that point
- a = - ω^2 x
SHM relationships
Displacement - time
(i) Start at x = 0 , with +v
- X = Xo sin ωt
(ii) Start at +Xo , with v = 0
- X = Xo cos ωt
(iii) Start at + X ' , with +v
- X = Xo sin ( ωt + Ф1 )
or
- X =Xo cos ( ωt + Ф2 )
Velocity - time
(i) X = Xo sin ωt
- V = ωXo cos ωt
- a = - ω^2Xo sin ωt
(ii) X = Xo cos ωt
- V = ωXo sin ωt
- a = - ω^2Xo cos ωt
Velocity - displacement
- v = +- ω √ ( (Xo)^2 - X^2)
Uniform Circular Motion
- Frequency of SHM = frequency of Circular Motion
- Amplitude = radisu of circle
Energy
- KE = 1/2 m ω^2 ( (Xo)^2 - X^2)
- Max KE = 1/2 m ω^2 ( Xo )^2
- PE = TE - KE
- PE = 1/2 m ω^2 x^2
-Max PE = 1/2m ω^2 (Xo)^2
Damped Oscillations
- Progressive decrease in amplitude
- Examples:
(i) Air resistance
- increase surface area
(ii) Fluid Viscosity
(iii) Eddy Current
- Extent of Damping
(i) Light
- Decrease with same proportion
- Stops only after a large number of osciallations
(ii) Critical
- Returns to equilibrium in minimum time
(iii) Heavy
- Returns to equilibrium after a long time
Resonance
- It is a phenomenon in which an oscillatory system reacts with maximum amplitude to an external periodic force whose driving frequency matches the natural frequency of the driven system
- As system gets more heavily damped,
(i) Amplitude decreases
(ii) Resonance peak broader
(iii) Resonance peak shifts left