Chapter 11 Oscillations and Waves

Oscilllations

Period

Frequency

Period and Frequency

The dynamics of SHM

Hooke’s Law

Enegy

Elastic Potential Energy

The Kinematics of SHM

Pendulumns

Oscillations

Any motion that regularly repeats is referred to as periodic or harmonic motion. Common examples include an object undergoing uniform circular motion, a mass oscillating on a spring and a pendulum. This type of motion can be characterized by its period or frequency.

Period

The time it takes an object to move through one full cycle of motion is called the period. For an object undergoing uniform circular motion, the period is the time it takes to make one revolution. For a mass on a spring on a pendulum. It is the time it takes to make a round trip (ie. the final position and velocity must be the same as the initial values). The period is denoted by T and is measured in seconds.

put the example here.

Frequency

Rather than timing one cycle to find the period, we can instead count the number of cycles that occur in one second. This is known as the frequency, denoted by f. The units of f are cycles per second, or hertz.

Now the first thing we notice is that period and frequency are reciprocals. After all, the period is “the number of seconds per cycle,” and the frequency is “the number of cycles per second.” So, we have these fundamental relationships:

Period and frequency

f=1/T and T=1/f

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The dynamics of SHM

Force

Let’s first describe the motion of the block attached to the spring from the point of view of the force it feels. The spring exerts a force on the bock that’s proportional to its displacement. If we call the equilibrium position x=0, then the force exerted by the spring is given by

Hooke’s Law

F=-kx

The proportionality constant, k, called the spring constant, tells us how strong the spring is; the greater the value of k, the stiffer (and stronger) the spring

page 382.

Energy

Unfortunately, knowing an equation for the force doesn’t allow us to solve directly for other things, such as the speed of the block at some later time or the work done by or against the spring: the force changes as the block moves, so acceleration is not uniform. However, there is a way to figure out these quantities by using energy. When we pull on the spring to get the oscillations started, we’re exerting a force over a distance; that is, we’re doing work. Because we’re doing work against the spring, the spring stores potential energy, called elastic potential energy. If we once again call the equilibrium position of the spring x=0, then the potential energy of a stretched or compressed spring is given by this equation:

elastic potential energy

PE_elastic=1/2kx²

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The Kinematics of SHM

Earlier it was mentioned that a mass oscillating on a spring exhibits “ideal” oscillatory motion, which is called simple harmonic motion, and that this motion is the result of Hooke’s Law (i.e. the restoring force is directly proportional to the distance from equilibrium). But what makes this motion different than non-ideal oscillations? It turns out(using calculus) that the frequency and period only depend on the spring constant, k, and the mass of the block, m.

look at the formula on page 385.

Pendulums

Besides the spring-block harmonic oscillator, there’s another oscillator that the MCAT will expect you to know about: the simple pendulum. If the connecting rod or string between the suspension point and the object at the end of a pendulum has negligible mass (so that all the mass is in the object at the end of the rod or string), and if there is no friction at the suspension point during oscillation, we say the pendulum is a simple pendulum.

The displacement of the mass is not taken as a distance from equilibrium (as in the spring-block case), but rather as the angle it makes with the vertical. The vertical (shown as a dashed line in the figure below) is the equilibrium position, θ=0. The restoring force here is gravity; specifically, it’s equal to mg sin θ, which is the component of the object’s weight in the direction toward equilibrium.

There is a picture on top of page 383

Strictly speaking a pendulum does not undergo simply harmonic motion because the restoring force is not proportional to the displacement (mg sin θ is not exactly proportional to θ). However, if the angle is small, then sin θ is approximately θ (in radians), so the restoring force can be approximated as mg θ, which is proportional to θ. In this case, we can treat the motion as simple harmonic, and the frequency and period are given by the following equations:

the equation is on page 383

Where l (the lowercase version of “L”) is the length of the pendulum and is g is the acceleration due to gravity. Obseve that in the case of simple harmonic motion of a simple pendulum, the mass of the swinging object does not affect the frequency or period of oscillation.

There is an example at the bottom of page 387.

Waves

A mechanical wave is a series of disturbances (i.e. oscillations) within a medium that transfers energy from one place to another. The medium itself is not transported, just the energy. Examples included a vibrating string or sound. Mechanical weaves cannot exist without a medium. In a later chapter we will discuss electromagnetic waves, which do not need a medium. This is because the electric and magnetic fields oscillate rather than physical matter.

Transverse waves

Perhaps the simplest example of wave is one we can create by wiggling one end of a long rope:

there is a picture on page 388

This wave uses the rope as the medium, traveling from one end to the other. Notice that the wave is moving horizontally, but the rope itself is moving up and down. That’s why this is called a transverse wave: the wave travels (propagates) in a direction that’s perpendicular to the direction in which the medium is vibrating.

Frequency and Period

The most fundamental characteristic of wave is its frequency. If we pick a spot on the rope and count how many times it moves up and down (the number of round trips it makes) in one second, we’ve just measured the frequency, f, which we express in hertz (cycles per second).

The period of a wave, T, is the reciprocal of the frequency, and is the amount of time it takes any spot on the rope to complete once cycle (in this case, one up-and-down round trip).

These definitions for frequency and period at same as for a mass on a spring or pendulum. Each particle of rope oscillates up and down with simple harmonic motion. However, can also think of the frequency and period of a wave in a different way. Instead of focusing on the oscillations, we can observe “pulses” moving the right. Frequency can be thought as the number of pulses that pass a given point per unit time and period is the time it takes between pulses.

wavelengths and amplitude

The figure below identifies the crests (peaks) and troughs of the wave. The distance from one crest to the next (i.e., the length of one cycle of the wave) is called the wavelength, denoted by λ, the Greek letter lambda. We can also measure the wavelength by measuring the distance from one trough to the next, or, in fact, between two consecutive corresponding points along the wave.

There is a picture on page 389

The amplitude of a wave, A, is the maximum displacement from equilibrium that any point in the medium makes as the wave goes by. In the case of a wave on a rope, the amplitude is the distance from the original horizontal position of the rope up to a crest; it’s also the distance fro the horizontal position down to a trough.

Wave speed

To figure out how fast the wave travels, we just notice that the wave travels a distance of λ in time T; that is, λ, is the length of one wave cycle, and T, the period, is the time required for one wave cycle to go by Since distance = rate x time, we get λ = vT, Solving this for v gives us λ (1/T)=v and since f=1/T, the equation becomes v=λf. This is the most important equation for waves and one of the most important equations for the MCAT.

Wave equation: v=λf

Two Big Rules for Waves

Notice that the second equation for the wave speed shows that v does not depend on f (or λ). While this may seem to contradict the first equation, v=λf, it really doesnt. The speed of the wave depends on the characteristics of the rope: how tense it is, and what it’s made of. We can wiggle the end at any frequency we want, and the speed of the wave we create will be a constant. However, because λf=v must always be true. A higher f will means a shorter λ (and a lower f will mean a longer λ). Thus, changing f doesn’t change v: It changes λ. This brings up our first big rule for waves:

Big rule 1: the speed of a wave is determined by the type of wave and the characteristics of the medium, not by the frequency.

the rest of page 390.

Interference of waves

When two or more waves are superimposed on each other, they will combine to form a single resultant wave. This is called interference. The amplitude of the resultant wave will depend on the amplitude of the combining waves and on how these waves travel relative to each other.

If crest meets crest, and trough meets trough, we say the waves are in phase with each other. Their amplitudes will add, and we say the waves interfere constructively. However, if the crest of one wave coincides with the trough of the other (and vice versa), we say that the waves are exactly out of phase with each other. In this case, their amplitudes subtract and we say that the waves interfere destructively.

the rest of page 393, including the picture.

the rest of page 394

Standing waves

Let’s say that we have a long rope with one end in our fingers and the other end attached to a wall. We wiggle the rope up and down at a certain frequency, f, and create waves of frequency f that travel down the length of the rope. When they hit the wall, they will be reflected. We now have two waves on the same rope (the wave we continue to generate plus the reflected wave) with the same frequency and amplitude but traveling in the opposite directions. These waves will interfere. If the frequency is just right, the resulting wave seems to stand still; the rope continues to vibrate up and down, but the resultant wave no longer travels. The combinations of these traveling waves produces a standing wave, with the horizontal positions of the crests and troughs remaining fixed.

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