Energy of the Simple Harmonic Oscillator

The Total Mechanical Energy (PE + KE) Is Constant KINETIC ENERGY: KE = mv2Remember v = -Asin(t+)KE = m2A2sin2(t+)

The Total Mechanical Energy (PE + KE) Is Constant POTENTIAL ENERGY: PE = kx2Remember x = Acos(t+)PE = kA2cos2(t+)

The Total Mechanical Energy (PE + KE) Is Constant Etot = KE + PEEtot = kA2(sin2(t+) + cos2(t+))Remember: 2 = k/msin2 + cos2 = 1Therefore Etot = kA2

Note that PE is small when KE is large and vice versaThe sum of PE and KE is constant and the sum = kA2Both PE and KE are always positivePE and KE vs time is shown on the leftThe variations of PE and KE with the displacement x are shown on the right

Velocity as a function of position for a Simple Harmonic Oscillator

The Simple Pendulum

The forces acting on the bob are tension, T, and the gravitational force, mg. The tangential component of the gravitational force, mgsin, always acts in the opposite direction of the displacement and is the restorative force.

Where s is the displacement along the arc and s=LmTmgmgsinmgcosL

The equation then reduces to:

But this is not of the form:

because the second derivative is proportional to sin, not mTmgmgsinmgcosL

BUT we can assume that if is small that sin= (this is called the small angle approximation)

So now the equation becomes:

And now the expression follows that for simple harmonic motion

mTmgmgsinmgcosL

SHM: The PendulumFrom this equation can be written as: = maxcos(t+)

max is the maximum angular displacement

, the angular frequency, is:

because this follows the function

The Period, T of the motion would be:

Damped OscillationsIn many cases dissipative forces (like friction) act on an object.The Mechanical Energy diminishes with time and the motion is dampedThe retarding force can be expressed as: R = -bv (b is a constant, the damping coefficient)The restoring force can be expressed as F = -kx

When we do the sum of the forces:

The solution to this equation follows the form:

Damped Oscillations

When the retarding force < the restoring force, the oscillatory character is preserved but the amplitude decreases

The amplitude decays exponentially with time

Damped Oscillations

You can also express as:o = (k/m)o is the natural frequency

Damped OscillationsWhen the magnitude of the maximum retarding force bvmax< kA, the system is underdampedWhen b reaches a critical value, bc= 2mo, the system does not oscillate and is critically dampedIf the retarding force is greater than the restoring force, bvmax > kA, the system is overdamped

Forced OscillationsThe amplitude will remain constant if the energy input per cycle equals the energy lost due to dampingThis type of motion is called a force oscillationThen the sum of the forces becomes:

Forced Oscillations

The solution to this equation follows the form:

Forced OscillationsWhen the frequency of the driving force equals the natural frequency o, resonance occurs

At resonance the applied force is in phase with the velocityAt resonance the power transferred to the oscillator is at a maximum

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