Energy of the Simple Harmonic Oscillator

The Total Mechanical Energy (PE + KE) Is Constant KINETIC ENERGY: KE = ½ mv 2 Remember v = -ωAsin(ωt+ ϕ ) KE = ½ mω 2 A 2 sin 2 (ωt+ ϕ )

The Total Mechanical Energy (PE + KE) Is Constant POTENTIAL ENERGY: PE = ½ kx 2 Remember x = Acos(ωt+ ϕ ) PE = ½ kA 2 cos 2 (ωt+ ϕ )

The Total Mechanical Energy (PE + KE) Is Constant E tot = KE + PE E tot = ½kA 2 (sin 2 (ωt+ ϕ ) + cos 2 (ωt+ ϕ )) Remember: ω 2 = k/m sin 2 θ + cos 2 θ = 1 Therefore E tot = ½kA 2

Note that PE is small when KE is large and vice versa The sum of PE and KE is constant and the sum = ½ kA 2 Both PE and KE are always positive PE and KE vs time is shown on the left The 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, mgsinθ, always acts in the opposite direction of the displacement and is the restorative force. Where s is the displacement along the arc and s=Lθ m T mg mgsinθ mgcosθ L θ

The equation then reduces to: But this is not of the form: because the second derivative is proportional to sinθ, not θ m T mg mgsinθ mgcosθ L θ

BUT… we can assume that if θ is small that sinθ=θ (this is called the small angle approximation) So now the equation becomes: And now the expression follows that for simple harmonic motion m T mg mgsinθ mgcosθ L θ

SHM: The Pendulum From this equation θ can be written as: θ = θ max cos(ωt+Φ) Θ max is the maximum angular displacement ω, the angular frequency, is: because this follows the function The Period, T of the motion would be:

Damped Oscillations In many cases dissipative forces (like friction) act on an object. The Mechanical Energy diminishes with time and the motion is damped The 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 ω as: ω o = √(k/m) ω o is the natural frequency Damped Oscillations When the magnitude of the maximum retarding force bv max < kA, the system is underdamped When b reaches a critical value, b c = 2mω o, the system does not oscillate and is critically damped If the retarding force is greater than the restoring force, bv max > kA, the system is overdamped

Forced Oscillations The amplitude will remain constant if the energy input per cycle equals the energy lost due to damping This type of motion is called a force oscillation Then the sum of the forces becomes:

Forced Oscillations The solution to this equation follows the form:

Forced Oscillations When the frequency of the driving force equals the natural frequency ω o, resonance occurs At resonance the applied force is in phase with the velocity At resonance the power transferred to the oscillator is at a maximum