Class 11 Physics Oscillations Damped Simple Harmonic Motion

DAMPED SIMPLE HARMONIC MOTION

Damped SHM can be stated as:-

1. Motion in which amplitude of the oscillating body reduces and eventually comes to its mean position.
2. Dissipating forces cause damping.
3. Consider a pendulum which is oscillating
4. After some time we can observe that its displacement starts decreasing and finally it comes to rest.
5. This implies that there is some resistive force which opposes the motion of the pendulum. This type of SHM is known as Damped SHM.

Damping Force:-

• It opposes the motion of thebody.
• Magnitude of damping force is proportional to the velocity of the body.
• It actsin the opposite direction of the velocity.
• Denoted by Fdwhere d is the damping force.
• Fd= -b v where b is a damping constant and it depends on characteristics of the medium (viscosity, for example) and the size and shape of the block.
• (-ive) directed opposite to velocity

Equation for Damped oscillations: Consider a pendulum which is oscillating.

It will experience two forces

1. Restoring force Fs = -k x
2. Damping Force Fd = -b v

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.

• The figure has 4 pendulums and the strings to which pendulum bobs 1 and 4 are attached are of the same length and the others are of different lengths.
• Once displaced, the energy from this pendulum gets transferred to other pendulums through the connecting rope and they start oscillating. The driving force is provided through the connecting rope and the frequency of this force is the same as that of pendulum 1.
• Once pendulum 1 is displaced, pendulums 2, 3 and 5 initially start oscillating with their natural frequenciesand different amplitudes, but this motion is gradually damped and not sustained.
• Their oscillation frequencies slowly change and later start oscillating with thefrequency of pendulum 1, i.e. the frequency ofdriving force but with different amplitudes.
• They oscillate with small amplitudes. The oscillation frequency of pendulum 4 is different than pendulums 2, 3 and 5.
• Pendulum 4 oscillates with the same frequency as that of pendulum 1 and its amplitude gradually picks up and becomes very large.
• This happens due to the condition for resonance getting satisfied, i.e. the natural frequency of the system coincides with that of the driving force

.