Unit 4: Oscillations

Table of Contents

Simple Harmonic Oscillations (SHM)

Simple Harmonic Motion (SHM) is a special type of periodic motion where the restoring force (F) is directly proportional to the displacement (x) from the equilibrium position and acts in the opposite direction.

This is Hooke's Law: F = -kx
Where `k` is the force constant (or spring constant).

Differential Equation of SHM

Using Newton's Second Law, F = ma:

ma = -kx

m(d²x/dt²) = -kx

m(d²x/dt²) + kx = 0

Dividing by `m`, we get the standard form:

d²x/dt² + (k/m)x = 0

We define the natural angular frequency (ω₀) as:

ω₀² = k/m (or ω₀ = √(k/m))

So the differential equation becomes:

d²x/dt² + ω₀²x = 0

Solution of the SHM Equation

This is a second-order homogeneous differential equation. The general solution can be written in several equivalent forms:

  1. Sine/Cosine Form: x(t) = B sin(ω₀t) + C cos(ω₀t)
  2. Amplitude/Phase Form: This is the most common form.
    x(t) = A cos(ω₀t + φ)
    • A = Amplitude: The maximum displacement from equilibrium.
    • ω₀ = Angular Frequency: (radians/sec). Related to frequency (f) and period (T) by `ω₀ = 2πf = 2π/T`.
    • (ω₀t + φ) = Phase: The argument of the cosine function.
    • φ = Phase Constant (or Phase Angle): Determined by the initial conditions (x and v at t=0). It tells you where in the cycle the motion starts.

Velocity and Acceleration in SHM:

Energy in SHM and Time Average Values

Kinetic Energy (K)

K = (1/2)mv² = (1/2)m [-Aω₀ sin(ω₀t + φ)]²

K(t) = (1/2)mω₀²A² sin²(ω₀t + φ)

Since ω₀² = k/m, this is also K(t) = (1/2)kA² sin²(ω₀t + φ).

Potential Energy (U)

For a spring system, U = (1/2)kx² = (1/2)k [A cos(ω₀t + φ)]²

U(t) = (1/2)kA² cos²(ω₀t + φ)

Total Energy (E)

E = K(t) + U(t) = (1/2)kA² sin²(ω₀t + φ) + (1/2)kA² cos²(ω₀t + φ)

Using the identity sin²θ + cos²θ = 1:

E = (1/2)kA² = constant

The total mechanical energy in an *undamped* simple harmonic oscillator is constant and is proportional to the square of the amplitude.

Time Average Values

The average value of a function `f(t)` over one period `T` is `(1/T) ∫[0, T] f(t) dt`.

The key trigonometric identity is that the average value of sin²(θ) or cos²(θ) over one full period is 1/2.

Average Kinetic Energy ()

= <(1/2)kA² sin²(ω₀t + φ)>

= (1/2)kA² * = (1/2)kA² * (1/2)

= (1/4)kA² = (1/2) E

Average Potential Energy ()

= <(1/2)kA² cos²(ω₀t + φ)>

= (1/2)kA² * = (1/2)kA² * (1/2)

= (1/4)kA² = (1/2) E

Conclusion: For SHM, the average kinetic energy is equal to the average potential energy, and each is equal to half the total energy.

Damped Oscillation

This is a more realistic model of an oscillator, which includes a damping force (like friction or air resistance) that opposes the motion. This force is typically modeled as being proportional to the velocity:

Fdamping = -bv (where `b` is the damping coefficient)

The total force is now Fnet = Frestoring + Fdamping = -kx - bv

The differential equation becomes:

m(d²x/dt²) = -kx - b(dx/dt)

m(d²x/dt²) + b(dx/dt) + kx = 0

We define the damping constant γ = b/m (some books use 2γ = b/m, be careful). Let's use 2β = b/m for simplicity in the solution (where β is the decay constant).

d²x/dt² + (b/m)(dx/dt) + (k/m)x = 0

d²x/dt² + 2β(dx/dt) + ω₀²x = 0 (where β = b/2m and ω₀² = k/m)

The solution to this equation is:

x(t) = A0 e-βt cos(ω't + φ)

This is an oscillation `cos(ω't + φ)` with an exponentially decaying amplitude `A(t) = A₀e⁻ᵇᵗ`.

The new angular frequency `ω'` is slower than the natural frequency `ω₀`:

ω' = √(ω₀² - β²) = √( (k/m) - (b/2m)² )

Three Cases of Damping:

  1. Underdamped (β < ω₀): This is the case above. The system oscillates with decreasing amplitude. `ω'` is real.
  2. Critically Damped (β = ω₀): The system returns to equilibrium as fast as possible without oscillating. (e.g., car shock absorbers).
  3. Overdamped (β > ω₀): The system returns to equilibrium slowly without oscillating (like a door closer in molasses). `ω'` is imaginary.

Forced Oscillations and Resonance

This is a damped oscillator that is also pushed by an external, periodic driving force. Let the driving force be `F_drive = F₀ cos(ωt)` (where `ω` is the driving frequency, which is *not* the same as `ω₀` or `ω'`).

The full differential equation is:

m(d²x/dt²) + b(dx/dt) + kx = F₀ cos(ωt)

The general solution is `x(t) = x_c(t) + x_p(t)`.

  • `x_c(t)` is the "transient" solution (the damped oscillation from before, `A₀e⁻ᵇᵗ...`). This part decays to zero over time.
  • `x_p(t)` is the "steady-state" solution. This part survives and describes the long-term motion.

The steady-state solution `x_p(t)` is an oscillation at the *driving frequency* `ω`, but with a phase shift `φ` relative to the driver:

x(t) = A(ω) cos(ωt - φ)

The Amplitude A(ω) of this steady-state motion depends on the driving frequency `ω`:

A(ω) = (F₀/m) / √[ (ω₀² - ω²)² + (bω/m)² ]

This amplitude `A(ω)` is small when `ω` is very low or very high. It becomes maximum when the driving frequency `ω` is *close* to the natural frequency `ω₀`.

Resonance

Resonance is the phenomenon where the amplitude of a forced oscillator becomes very large when the driving frequency (ω) is tuned to a specific value, called the resonance frequency (ωR).

The resonance frequency is:

ωR = √(ω₀² - 2β²) = √( (k/m) - b²/2m² )

Key Point:
  • If damping `b` is very small, then ωR ≈ ω₀.
  • Resonance occurs when you drive the system at its natural frequency.
  • The amplitude at resonance is `A_max ≈ F₀ / (bω₀)`. If `b=0` (no damping), the amplitude theoretically goes to infinity.
Examples: Pushing a child on a swing (you push at the swing's natural frequency), shattering a wine glass with a specific sound frequency, Tacoma Narrows Bridge collapse.

Sharpness of Resonance

The "sharpness" describes how narrow the resonance peak is.

  • Low Damping (small b): Very large amplitude, but only for frequencies *very* close to `ω₀`. This is a sharp resonance.
  • High Damping (large b): Smaller amplitude, but the response is "broad" over a wider range of frequencies.

Power Dissipation and Quality Factor (Q)

Power Dissipation

In a damped (or forced) oscillator, energy is continuously lost (dissipated) by the damping force `F_damping = -bv`.
The instantaneous power dissipated is P(t) = F_damping * v = (-bv) * v = -bv².

The average power dissipated `

` in the steady-state of a forced oscillator is:

= (1/2) b ω² A(ω)²

This power is supplied by the driving force. At resonance, the amplitude `A` is max, so the power transfer from the driver to the oscillator is also maximum.

Quality Factor (Q)

The Q Factor is a dimensionless parameter that describes how "good" an oscillator is (how low its damping is). A high Q-factor means low damping.

It can be defined in several equivalent ways:

  1. Definition 1 (Energy): Based on energy loss per cycle.
    Q = 2π * (Energy Stored in Oscillator / Energy Lost per Cycle)
  2. Definition 2 (Damping): Based on the system parameters.
    Q = ω₀ / (b/m) = ω₀ / (2β)
  3. Definition 3 (Resonance Sharpness): Based on the resonance curve.
    Let Δω be the "full width at half maximum" (FWHM) of the *energy* resonance peak.
    Q ≈ ω₀ / Δω
    A high Q-factor means a small Δω, which is a very sharp resonance.

Non-Inertial Systems and Fictitious Forces

(This topic was introduced in Unit 1 and is expanded here.)

Recall: A non-inertial frame is an accelerating frame of reference. In such a frame, Newton's 1st Law (inertia) fails. Objects appear to accelerate without a real force.

To "fix" Newton's 2nd Law (F=ma) so we can still use it in a non-inertial frame, we must invent fictitious forces (or pseudo-forces). These are not real forces (not part of a 3rd Law pair) but are mathematical corrections due to the frame's acceleration.

Let `a_inertial` be the true acceleration of an object in an inertial frame, and `a_non_inertial` be its acceleration *as measured by an observer* in the non-inertial frame.
Let `A` be the acceleration of the non-inertial frame itself (relative to the inertial frame).

The relationship is: ainertial = anon_inertial + A

Now, Newton's 2nd Law (which is only valid in the inertial frame) is:

Freal = m ainertial = m(anon_inertial + A)

Freal = m anon_inertial + mA

Rearranging this to look like F=ma *in the non-inertial frame*:

m anon_inertial = Freal - mA

We define the fictitious force (Ffict) as:

Ffict = -m A

So, the "law" in the non-inertial frame becomes:

m anon_inertial = Freal + Ffict

Example: A person of mass `m` in an elevator accelerating upwards with `A = +aĵ`.

  • The real forces are gravity (Fg = -mgĵ) and the normal force (FN = FNĵ).
  • The person is not accelerating *relative to the frame* (anon_inertial = 0).
  • The fictitious force is Ffict = -m(aĵ) = -maĵ (a "force" pushing them down).
  • Applying the new law: m(0) = Freal + Ffict = (FNĵ - mgĵ) + (-maĵ)
  • 0 = FN - mg - ma → FN = mg + ma. The person "feels heavier".

Uniformly Rotating Frame: Coriolis and Centrifugal Force

A uniformly rotating frame (like the Earth) is a non-inertial frame. The full fictitious force equation for a rotating frame is more complex. It has two parts:

Ffict = Fcentrifugal + FCoriolis

Centrifugal Force

This is the familiar "force" that seems to "fling" objects outward from the center of rotation.

Fcentrifugal = -m (ω × (ω × r))

Magnitude: For circular motion, this simplifies to Fcentrifugal = +mω²r, directed radially outward.

This is the force you "feel" on a merry-go-round. It's just your body's inertia (its desire to go in a straight line) being interpreted as an outward force in the rotating frame.

Coriolis Force

This is a more subtle fictitious force that acts only on objects that are moving relative to the rotating frame.

FCoriolis = -2m (ω × vnon_inertial)

Where `ω` is the angular velocity vector of the frame and `v_non_inertial` is the object's velocity *as measured in the rotating frame*.

Characteristics:

  • The force is perpendicular to both the axis of rotation (`ω`) and the object's velocity (`v`).
  • It does no work (since F is perpendicular to v).
  • It acts as a deflecting force.
Applications (Coriolis Effect on Earth):
  • Earth's `ω` vector points North (out of the North Pole).
  • Northern Hemisphere: Deflects moving objects to the right.
    • This is why hurricanes and cyclones spin counter-clockwise (air rushing *in* is deflected right, setting up the spin).
    • Causes long-range artillery shells to drift right.
  • Southern Hemisphere: Deflects moving objects to the left.
    • Cyclones spin clockwise.
  • At the Equator: `ω` is horizontal. The Coriolis force is vertical (for horizontal motion) and has no effect on cyclone formation. This is why hurricanes do not form at the equator.