### Unit 1: Mechanical Vibrations and Waves in Extended Objects

The material in this unit is not directly related to electricity and magnetism, but it is the foundation of one of the most significant outcomes of Maxwell's theory – electromagnetic waves. In PHYS101: Introduction to Mechanics, we learned how to describe the motion of particle-like masses using classical mechanics. In this unit, we begin the transition from mechanics to electromagnetism by examining how objects of size – length, width, and depth – behave.

When we look at these types of extended objects, a mystery is hiding in plain sight: if you pull one end of a rope, how does the other end know? Your action somehow "propagates" from one end to the other. The answer is related to the invisible hand of electromagnetism that can transmit information between different locations.

In this unit, we focus on vibrating systems and the propagation of mechanical waves through media; think of ripples traveling outward from a stone dropped into water. We also lay the basic foundation for the development of a classical theory of mechanics for extended solids.

**Completing this unit should take you approximately 7 hours.**

Upon successful completion of this unit, you will be able to:

- describe the properties of simple harmonic motion and provide examples;
- define the following terms related to wave motion: frequency, wavelength, diffraction, and interference;
- state Hooke's Law; and
- solve problems using simple harmonic motion.

### 1.1: Periodic Motion and Simple Harmonic Oscillators

Watch these videos.

This demonstration illustrates a very simple example of a non-harmonic oscillator: a helium balloon on a string. There are two sources of non-linearity. First, as the balloon rises, it lifts more string. Therefore, the mass of the oscillator is a function of the position of the oscillating mass, which leads to nonlinear behavior. A damping term has also been included, which mimics the effect of air resistance. Adjust the various control parameters to gain a feel for which parameters have a larger effect on the motion of the oscillator. Check out the special cases suggested in the "Details" section of the demonstration: motion in the absence of damping (set the damping constant to 0), motion of the balloon when the string has no mass (the force of gravity no longer increases as the balloon goes up), and motion when the mass of the string is large and the damping constant is large. What happens to the balloon eventually in this case?

Read this chapter and try each of the six problems before looking at the solutions. Make sure you understand not only the solutions but also how to approach solving the problem so that you can obtain the solution yourself. You will solve these kinds of problems on the final exam.

This demonstration illustrates the motion of a mass on a spring. When the mass is pulled down, the spring exerts a restoring force described by Hooke's Law that pulls the mass upwards. The result is that the mass travels up and down in simple harmonic motion, where the displacement of the mass is described by a sinusoidal curve. Think of this demonstration as an experiment to verify (or not) the effect of Hooke's Law on the period of oscillation. Use this worksheet as your guide in working with this demonstration.

### 1.2: Vibrations

Download this book. This is a large file, but we will use it throughout the course. This version contains the solutions to the Self-Check questions. Read sections 1 and 2 of "Chapter 17: Vibrations" on pages 445–459.

### 1.3: Wave Motion

Watch this lecture series.

Read "Chapter 19: Free Waves" on pages 481–499. Answer the self-check questions. You can find the answers on page 553. Think about the discussion questions and solve problems 1–4 on pages 507–508. You can check some of the answers here.

This demonstration illustrates the superposition of two waves traveling in opposite directions. First, try setting the frequencies of the two waves to be equal. Notice that as the time passes, the superposition of two waves goes from "double" the wave (the wave with the same frequency and twice the amplitude), when the waves are in the same phase, to "no wave" (the waves cancel each other out completely) when they are in the opposite phase. Then, explore what happens when the frequencies of the waves are close to each other, but a little bit different. How does the superimposed wave look like? This effect is easier to see when the frequencies are large. Try clicking on the "plus" icon in the top right corner of the demonstration, and selecting "autorun". Notice that the superimposed wave contains an oscillation within an oscillation, one with the sum, and another one with the difference of the original frequencies.

Waves are partially reflected by local changes in the medium through which they propagate. This is illustrated here by the introduction of a point mass on a vibrating string. The transmitted wave becomes smaller in amplitude as the mass becomes larger. Why? Is there a phase shift associated with reflection/transmission? Why?

### Unit 1 Assessment

Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment