Physics 238—Intermediate Physics Lab Homework Assignment #4 Due Monday, April 21, 2025, 1:15 p.m.

In this assignment, you will explore the relation between the width of a resonance peak and the lifetime of a damped oscillator.

1 Background Theory

Damped Oscillations To start, consider the decaying oscillations of a damped torsional oscillator. The differential equation for a damped, torsional oscillator of torsional constant κ and rotational inertia I is

θ̈  = −ω_{0}^{2}θ − γθ̇

where ω_{0} = 2πf_{0} = \sqrt{\frac{κ}{I}} is the natural (undamped) frequency of the oscillator, and γ is a term proportional to the drag, with units of rad s^{−1}. If released from rest from an initial angle θ_{0}, the resulting motion is a damped oscillation:

θ(t)  = θ_{0} e^{−γt/2} cos(ω_{v}t + ϕ) .

For this assignment, the key observation is to note that the amplitude decays exponentially:

A  = θ_{0}e^{−γt/2} = θ_{0}e^{−t/(2τ)}

where τ = 1/γ is the time constant, or lifetime, associated with the motion. Recall that the energy of a simple harmonic oscillator is proportional to the amplitude squared, so we can also describe the decay in terms of energy:

A^{2}  = E = E_{0}e^{−γt} = E_{0}e^{−t/τ}  (1)

Driven Oscillations If periodic forcing at frequency ω_{d} is added, the new differential equation is

θ̈  = −ω_{0}^{2}θ − γθ̇ + α_{0} sin (ω_{d}t)

where α_{0} = \frac{τ_{0}}{I} is the amplitude of the driving. After an initial transient, the steady state motion is given by a sinusoidal oscillation with an amplitude θ_{0} given by

θ_{0}  = \frac{α_{0}}{\sqrt{(ω_{0}^{2} − ω_{d}^{2})^{2} + (γω_{d})^{2}}}

As with the decaying oscillation, the energy is related to the amplitude squared

θ_{0}^{2}  = \frac{α_{0}^{2}}{(ω_{0}^{2} − ω_{d}^{2})^{2} + (γω_{d})^{2}}  (2)

In this assignment, you will explore the relation between the lifetime τ of the damped oscillations and the width of the peak in Eq.x2. Throughout, it will be simpler to leave things in terms of ω and γ (or τ), rather than using f and Q.

Hint: Assume that the damping is small, so that γ ≪ ω_{0} (or equivalently Q ≫ 1) so that you may drop higher order terms in γ/ω_{0}, and use the binomial expansion when appropriate.

Assignment 1: What frequency ω_{d} gives the maximum for θ_{0}^{2} in Eq.x2? Call that frequency ω_{p}. What is the magnitude of θ_{0}^{2} at that frequency?

Assignment 2: What are the two frequencies that give an amplitude in Eq.x2 equal to one half of that maximum? Only keep terms up to the lowest order in γ/ω_{0}.

Assignment 3: Define the full width at half maximum—FWHM—as the difference between the two frequencies you found in assignment 2. Find the relation between the FWHM and τ in Eq.x1.

Assignment 4: Test your prediction for your mechanical resonance curve for the torsional oscillator. Note that since you measured amplitude θ and not θ^{2}, you will measure the FWHM by noting the frequencies at which your curve is at a height of \frac{1}{\sqrt{2}} of the maximum. Note too that you measured linear frequencies f, not angular frequencies ω, so you will need to manually handle the factor of 2π.

Assignment 5: Test your prediction for your electrical resonance curve for the RLC oscillator. Again, you will actually need to measure the frequencies at which your curve is at a height of \frac{1}{\sqrt{2}} of the maximum, and compensate for factors of 2π.

Conclusion Your result should illustrate a quite general relation between the width of a resonance curve and the lifetime of the corresponding state.