Day 25 - Resonance

Announcements

• Midterm 1 is still being graded
• Homework 6 is due Friday
• Homework 7 is posted, due next Friday
• No office hours today

Seminars this week

Most of MSU folks are at APS Global Physics Summit

WEDNESDAY, March 19, 2025

• Astronomy Seminar, 1:30 pm, 1400 BPS, Alex Rodriguez, University of Michigan, Galaxy clusters, cosmology, and velocity dispersion
• FRIB Nuclear Science Seminar, 3:30pm., FRIB 1300 Auditorium, Dr. Pierre Morfouace of CEA-DAM, Mapping the new asymmetric fission island with the R3B/SOFIA setup

THURSDAY, March 20, 2025

Colloquium, 3:30 pm, 1415 BPS, Guillaume Pignol, University of Grenoble, Ultracold neutrons: a precision tool in fundamental physics

Reminders

We started to solve the forced harmonic oscillator equation:

We examined the case of a sinusoidal driving force:

There's a complimentary case where the driving force is a sine wave:

Reminders

We combined the two equations into a complex equation using these identities:

The resulting equation is:

Notice that there's a homogeneous part () and a particular part ().

Reminders

The homogeneous part is the solution we've found before with the general solution:

where . In the case of a weakly damped oscillator (), we have:

These solutions die out as . They are called transient solutions.

Solving the particular part

The particular part is the solution to the driven harmonic oscillator equation:

Assume a sinusoidal solution (frequency, ) of the form:

where is a complex number. Then, we have:

Amplitude of the particular solution

We want to convert this to polar form:

where and are real numbers. We use the complex form to compute the magnitude of the amplitude:

Clicker Question 24-5

We found that the square amplitude of the driven harmonic oscillator is:

When is the amplitude of the driven oscillator maximized?

1. When the driving frequency () is far from the natural frequency ()
2. When the driving frequency () is close to the natural frequency ()
3. When the damping () is weak
4. When the damping () is strong
5. Some combination of the above

Finding the phase

With,

then we can compare the complex forms:

Both and are real numbers, so the phase is the same phase as the complex number:

The Particular Solution

Let's return to the particular solution:

So we get solutions to both driven oscillators:

These are the steady-state solutions.

They persist as and oscillate at the driving frequency .

The Full Solution

Here, is the transient solution and is the steady-state solution.

For weakly damped oscillators, the transient solution can be written in the form:

where and are real numbers and are the amplitude and phase of the transient solution. Both are determined by the initial conditions.

where and are real numbers and are the amplitude and phase of the steady-state solution.

The Full Solution

The transient plus the steady-state solution is the full solution:

As , the transient solution dies out and the steady-state solution persists.

where

Resonance

The amplitude of the steady-state solution is:

We change and observe how the amplitude changes.

Achieving resonance

The denominator of the equation controls the amplitude:

Case 1: Tune to be close to . Car Radio tuning

With , the amplitude is:

Achieving resonance

The denominator of the equation controls the amplitude:

Case 2: Tune to be close to . Pushing a swing

Find the that maximizes the amplitude by taking the derivative with respect to :