|Class 11 Physics Oscillations||Damped Simple Harmonic Motion|
DAMPED SIMPLE HARMONIC MOTION
Damped SHM can be stated as:-
Equation for Damped oscillations: Consider a pendulum which is oscillating.
It will experience two forces
The total force Ftotal = Fs + Fd = -k x – b v
Let a (t) = acceleration of the block
Ftotal= m a (t)
-k x – b v = md2x/dt2
md2x/dt2 + kx + bv =0
or md2x/dt2 + b dx/dt+ kx=0 (v=dx/dt) (differential equation)
d2x/dt2+ (b/m) dx/dt+ (k/m) x=0
After solving this equation
x(t) = A e–b t/2m cos (ω′t + φ ) (Equation of damped oscillations)
Damping is caused by the term e–b t/2m
ω’ =angular frequency
Mathematically can be given as:-
ω ′= −√ (k/m –b2/4m2)
Consider if b=0 (where b= damping force) then
x (t) = cos (ω′t + φ)( Equation of Simple Harmonic motion)
Graphically if we plot Damped Oscillations
There is exponentially decrease in amplitude with time.
Free Oscillations: - In these types of oscillations the amplitude and time period remain constant it does not change. This means there is no damping. But in real scenario there is no system which has constant amplitude and time period.
Forced Oscillations: - If weapply some external force to keep oscillations continue such oscillations are known as forced oscillations.In forced oscillations the system oscillates not with natural frequency but with the external frequency.
Example of Forced oscillation is when a child in a garden swing periodically presses his feet against the ground (or someone else periodically gives the child a push) to maintain the oscillations.
Resonance: -The phenomenonof increase in amplitude when the driving force
is close to the natural frequency of the oscillatoris called resonance. If an external force with angular frequency ωd acts on an oscillating system with natural angular frequency ω, the system oscillates with angular frequency ωd. The amplitude of oscillations is the greatest when
ωd = ω
thisexpression is called resonance. Swings are very good example of resonance.
Pic: Child swinging on the swing
In the above figure there are set of 5 pendulums of different lengths suspended from a common rope.