Energy of the Simple Harmonic Oscillator

The Total Mechanical Energy (PE + KE) Is

Constant

KINETIC ENERGY:

•KE = ½ mv2

•Remember v = -ωAsin(ωt+ϕ)

•KE = ½ mω2A2sin2(ωt+ϕ)

The Total Mechanical Energy (PE + KE) Is

Constant

POTENTIAL ENERGY:

•PE = ½ kx2

•Remember x = Acos(ωt+ϕ)

•PE = ½ kA2cos2(ωt+ϕ)

The Total Mechanical Energy (PE + KE) Is

Constant Etot = KE + PE

•Etot = ½kA2(sin2(ωt+ϕ) + cos2(ωt+ϕ))

•Remember:

• ω2 = k/m

• sin2θ + cos2θ = 1

•Therefore Etot = ½kA2

• Note that PE is small when KE is large and vice versa

• The sum of PE and KE is constant and the sum = ½ kA2

• 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

€

E =KE + PE = 12mv

2 + 12 kx

2 = 12 kA

2

v =k

m(A2 − x 2) = ±ω A2 − x 2

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

€

Ft = −mgsinθ = ma = md2s

dt 2∑

m

T

mg

mgsinθmgcosθ

L

θ

• The equation then reduces to:

•

• But this is not of the form:

because the second derivative is proportional to sinθ, not θ

€

d2θ

dt 2= −

g

Lsinθ

m

T

mg

mgsinθmgcosθ

L

θ

€

d2x

dt 2= −ω 2x

• 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

mgsinθmgcosθ

L

θ

€

d2θ

dt 2= −

g

Lθ

SHM: The Pendulum• From this equation θ can be written as:

• θ = θmaxcos(ωt+Φ)

• Θmax is the maximum angular displacement

• ω, the angular frequency, is:

• because this follows the function

• The Period, T of the motion would be:

€

ω =g

L

€

d2x

dt 2= −ω 2x

€

T =2π

ω= 2π

L

g

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

€

Fx = −kx −bv = max∑

−kx −bdx

dt= m

d2x

dt 2

€

x = Ae− b2m t cos(ωt +φ)

ω =k

m−b

2m

⎛

⎝ ⎜

⎞

⎠ ⎟2

• 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:• ωo = √(k/m)

• ωo is the natural frequency

Damped Oscillations

€

ω = ω o2 −

b

2m

⎛

⎝ ⎜

⎞

⎠ ⎟2

•When the magnitude of the maximum retarding force bvmax< kA, the system is underdamped

•When b reaches a critical value, bc= 2mωo, the system does not oscillate and is critically damped

•If the retarding force is greater than the restoring force, bvmax > 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:

€

Fx = Fext cosωt − kx −bv = max∑

Forced Oscillations

• The solution to this equation follows the form:

€

Fext cosωt − kx −bdx

dt= m

d2x

dt 2

€

x = Acos(ωt +φ)

where :

A =Fext

m (ω 2 −ωo2)2 +

bω

m

⎛

⎝ ⎜

⎞

⎠ ⎟2

Forced Oscillations

• When the frequency of the driving force equals the natural frequency ωo, 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