PHYS102: Introduction to Electromagnetism

1a. State Hooke's Law

• How can Hooke's Law be used to analyze the motion of a system?
• How can the magnitude and direction of a restoring force be found?

Simple harmonic motion (SHM) arises when a restoring force appears in a system when the system is disturbed from equilibrium, and when the force is proportional to and has the opposite direction of the displacement. A typical example of this type of system is a point mass attached to the end of a massless spring. In this case, Hooke's Law states that the restoring force is $F_{x}=-kx$, where $x$ is the displacement of the mass and is equal to the length of the spring is stretched or compressed. $k$ is the spring constant, determined by the material of the spring and how tightly it is wound. The negative sign indicates the force is in the direction opposite to the displacement.

Review the basic behavior of springs with some examples in Spring Force.

1b. Describe the properties of simple harmonic motion and provide examples

• What kind of force causes the objects to undergo simple harmonic motion?
• How do position, velocity, and acceleration depend on time in SHM? What is their relationship?

Simple harmonic motion (SHM) is a unique kind of motion in which the position is a sinusoidal function of time:

$x=x_{max}sin(2\pi ft)$.

Here, $x_{max}$ is the amplitude, and $f$ is the frequency of the oscillations. We can calculate the period of the oscillations as $T=\frac{1}{f}$. In general, $x$ does not have to be a position; it could be any variable describing a system, such as the angle of a pendulum.

The reason we begin the course with this topic is that it introduces abstract concepts such as frequency, and later energy, in a context that is more tangible than the phenomena of electricity and magnetism that make up the bulk of the material you will encounter later. In mechanical oscillations, you can see the effect of the restoring force directly by measuring the acceleration of the oscillating object. Oscillation is a repeating type of motion where the velocity constantly changes in magnitude and even in direction. That is acceleration.

Whenever the oscillating object reaches its largest deviation from equilibrium, it feels the largest force and therefore the largest acceleration. When the object returns to its equilibrium position, it feels no force and therefore has no acceleration. But without a force, there's nothing to bring the mass to a halt, and so it overshoots its equilibrium and keeps going. To explain oscillation, it is important to understand the difference between velocity and acceleration: you can have a large velocity with zero acceleration, but you need a large acceleration to change velocity rapidly.

Review simple harmonic motion in The Simple Pendulum. Review the mathematical description and properties of simple harmonic motion as applied to a system of a mass on a spring in Simple Harmonic Motion: A Special Periodic Motion.

1c. Describe the forms energy is contained in oscillatory motion

• Which two forms of energy are in a continuous interplay when a harmonic oscillation takes place?
• At what stage of an oscillation does an object have its largest potential energy?
• When an object oscillates, at what points does its velocity change direction? When is its speed largest?

Energy is an overarching principle in physics because we can use it to make predictions about many different types of motion. The main ingredient needed to make the concept of energy useful in oscillatory motion is the potential energy which depends on the deviation from equilibrium and on the spring constant. When you add potential energy to the kinetic energy of the oscillating mass, their sum stays constant over time, in the ideal case of simple harmonic motion. This total amount of energy depends only on how the oscillation was originally launched. Knowing this, we can predict the speed of the oscillating object just based on how far away it happens to be from equilibrium.

To check your understanding, identify which of the pictures in Fig. 16.16 (a) – (d) in Energy and the Simple Harmonic Oscillator show the maximum potential energy. Hint: there are two of them, and they correspond to zero velocity.

1d. Define resonance

• Does the total potential and kinetic energy of a pendulum stay constant over time if the motion is damped?
• To harvest almonds from a tree, farmers use a machine that shakes the tree back and forth. This works best if the shaking is not too slow and not too fast. Why?
• What determines the resonance frequency of an oscillator that is driven by a periodic external force?

When you leave an oscillating mass alone, it shows a specific, "natural" frequency at which it likes to oscillate. Think of a vibrating guitar string that produces a certain pitch no matter how you pluck it. This natural frequency is also the frequency when it becomes easiest to supply external energy to the oscillating object. The reason you need an external force in the first place is that all realistic oscillations lose energy over time, which is called damping.

An example is how you have to push a playground swing at just the right rate to match the rate at which it would swing by itself, to keep it going. This phenomenon is called resonance, and the optimal frequency for energy transfer is the resonance frequency. You could also try to push the oscillator at a different frequency, but that would lead to less transferred energy per second. You can see the effect of the transferred energy in the amplitude of the oscillator. The amplitude is largest at the resonance frequency, and smaller if the driving frequency is too low or too high.

Be sure to understand the horizontal axis of Fig. 16.27 in Forced Oscillations and Resonance. It is the frequency with which the external force changes. The peak in the figure corresponds to the natural frequency, $f_{0}$.

1e. Define terms related to wave motion: frequency, wavelength, diffraction, and interference

• What are the properties of wave motion?
• How is wave motion different from particle motion?
• What is superposition?
• What is the relationship between frequency, wavelength, and velocity in a periodic wave?

Waves surrounding us have so many different types of manifestations – ripples on a surface of water, sound, and light – that many do not realize they all have the same underlying properties. The main property that distinguishes wave motion from the motion of matter particles is that waves can pass through each other without affecting each other's motion. Waves combine when more than one wave is present at the same place. We call this interference of waves superposition.

We characterize periodic waves by their wavelength – the distance the wave travels during one period (the time it takes for the oscillation to go through a full cycle). The relationship between the wavelength and period is $\lambda=vT$, where $v$ is the speed of the wave propagation. Alternatively, wavelength can be related to frequency. Frequency is the reciprocal of period, and is the number of cycles the wave goes through in one second: $f=\frac {1}{\tau}$. Thus, $\lambda = \frac {v}{\tau}$.

Review Superposition and Interference for more on waves. You can interpret the example of standing waves on a string as the interference of a traveling wave: another traveling wave is reflected at one end, and then propagates in the opposite direction.

1f. Solve problems using simple harmonic motion

• What are the necessary conditions for a system to undergo simple harmonic motion?
• What quantities are necessary to calculate the angular frequency, frequency, and period of oscillations?
• What quantities are necessary to calculate the total energy of oscillations at a given time?
• What is the relationship between displacement, velocity, and acceleration of a system at a given time? What is the relationship between the maximum values of these variables?

Many application problems that involve simple harmonic motion focus on the relationship between the maximum displacement (amplitude) and maximum velocity of a system. Many of these questions ask you to calculate the frequency, period, and energy of oscillations.

Any harmonic oscillator has a characteristic constant $k$ that we usually call the "spring constant" or stiffness. This is the proportionality constant between the restoring force and the displacement from equilibrium according to Hooke's Law, $F=-kx$. We can calculate the oscillation frequency $f$ from this spring constant and the mass of the object that is moving back and forth, according to the formula:

$f=\frac{1}{2 \pi}\sqrt{\frac{k}{m}}$.

You can also solve this for the spring constant:

$k=m(d\pi f)^{2}$

This allows you to find the spring constant if you know the mass and the oscillation frequency $f$.

Let's say you would like to know the acceleration of the oscillating mass at a given displacement $x$, but you are only given the frequency and not the spring constant. You can calculate the answer by inserting the last equation into Hooke's Law and combining it with Newton's Second Law, $f=ma$. The result is:

$a=-(2\pi f)^{2}x$

Review the basic formula for the oscillation frequency with an example in Simple Harmonic Motion: A Special Periodic Motion.

Unit 1 Vocabulary

You should be familiar with these terms to complete the final exam.

• acceleration
• amplitude
• cycle
• displacement
• energy (potential and kinetic)
• equilibrium
• force constant (same as spring constant)
• frequency
• Hooke's Law
• oscillation
• natural frequency
• period
• phase
• resonance
• restorative (or restoring) force
• simple harmonic motion
• superposition
• velocity
• vibration
• wave
• wavelength