Vibrations & Waves Unit 9

All around us we see things that wiggle and jiggle All around us we see things that wiggle and jiggle. A wiggle in time is a vibration. A vibration cannot exist in one instant, but needs time to move back and forth. A wiggle in space and time is a wave. A wave cannot exist in one place but must extend from one place to another.

Galileo discovered that the time a pendulum takes to swing back and forth through small angles does not depend on the mass of the pendulum or on the distance through which it swings. The time of a back and forth swing- called a period- depends only on the length of the pendulum and the acceleration of gravity.

Definitions Simple harmonic motion: the back and forth vibratory motion of a swinging pendulum Sine curve: pictorial representation of a wave Crest: high point of wave Trough: low point of wave Amplitude: distance from the mid-point to the crest (or trough) of the wave Frequency: how often a vibration occurs

The source of all waves is something that vibrates The source of all waves is something that vibrates. The frequency of the vibrating source and the frequency of the wave it produces are the same. If the frequency of a vibrating object is known, the period can be calculated. Frequency= ___1____ period The unit of frequency is the Hertz The unit of period is the second.

Most of the information around us gets to us in some form of wave Most of the information around us gets to us in some form of wave. Sound is energy that travels to our ears in the form of one kind of wave. Light is energy that comes to our eyes in the form of a different kind of wave (an electromagnetic wave). The signals that reach our radio and television sets also travel in the form of electromagnetic waves.

When energy is transferred by a wave from vibrating source to a distant receiver, there is no transfer of matter between the two points. When a string is shaken up and down, a disturbance moves along the length of the string. The string itself is not moving. Drop a stone in a pond and you’ll produce a wave that moves out from the center in an expanding circle. It is the disturbance that moves, not the water, for after the disturbance passes, the water is where it was before the wave passed.

The speed of a wave depends on the medium through which the wave moves The speed of a wave depends on the medium through which the wave moves. Sound waves, for example, move at speeds of about 340m/s in air (depending on temperature), and about four times faster in water. Whatever the medium, the speed, wavelength, and frequency of the wave are related. Wave speed= wavelength x frequency

Types of waves Transverse: The motion of the wave is at right angles to the direction in which the wave is moving Waves in the stretched of musical instruments and upon the surfaces of liquids are transverse, as well as the electromagnetic waves that make up radio waves and light. Longitudinal: Particles of the medium move along the direction of the wave Sound waves are longitudinal waves.

More than one wave can exist at the same time in the same space More than one wave can exist at the same time in the same space. If you drop two rocks in water, the waves produced by each can overlap and form an interference pattern. Within the pattern, waves may be increased, decreased, or neutralized.

When the crest of one wave overlaps the crest of another, their individual effects add together. The result is a wave of increased amplitude. This is called constructive interference, or reinforcement. When the crest of one wave overlaps the trough of another, their individual effects are reduced. The high part of one wave simply fills in the low part of another. This is called destructive interference, or cancellation. Interference is characteristic of all wave motion.

If you tie a rope to a wall and shake the free end up and down, you will produce a wave in the rope. By shaking the rope just right, you can cause the original and reflected waves to form a standing wave. In a standing wave certain parts of the rope, called the nodes, remain stationary. The positions on a standing wave with the largest amplitudes are known as antinodes.

Standing waves are a result of interference Standing waves are a result of interference. When two waves of equal amplitude and wavelength pass through each other in opposite directions, the waves are always out of phase at the nodes. The nodes are stable regions of destructive interference.

The apparent change in frequency due to the motion of the source (or receiver) is called the Doppler effect (after the Austrian scientist Christian Doppler, 1803-1853). The greater the speed of the source, the greater will be the Doppler effect. The Doppler effect is evident when you hear the changing pitch of a car horn as the car passes you. When the car approaches, the pitch sounds higher than normal. This occurs because the sound wave crests are encountering you more frequently. When the car passes and moves away, you hear a drop in pitch because the wave crests are encountering you less frequently.

Police make use of the Doppler effect of radar waves in measuring the speeds of cars on the highway. Radar waves are electromagnetic waves, lower in frequency than light and higher in frequency than radio waves. Police bounce them off moving cars, and a computer built into the radar system calculates the speed of the car relative to the radar unit by comparing the frequency of the radar emitted by the antenna with the frequency of the reflected waves.

The Doppler effect also occurs for light The Doppler effect also occurs for light. When a light source approaches, there is an increase in its measured frequency, and when it recedes, there is a decrease in its frequency. An increase in frequency is called a blue shift, because the increase is towards the high-frequency, or blue, end of the color spectrum. A decrease in frequency is called a red shift, referring to the low-frequency, or red, end of the color spectrum.

When the speed of the source in a medium is as great as the speed of the waves it produces, something interesting happens. The waves pile up. The same thing happens when an aircraft travels at the speed of sound. The overlapping wave crests disrupt the flow of air over the wings making it harder to control the plane when it is flying close to the speed of sound. An airplane with sufficient power, however, can easily travel faster than the speed of sound. Then we say that it is supersonic- faster than sound. A supersonic airplane flies into smooth, undisturbed air because no sound wave can propagate out in front of it. It outruns the waves it produces. The crests overlap at the edges and the pattern made by these overlapping crests is a V shape, called a bow wave, which appears to be dragging behind.

A speedboat knifing through the water generates a two-dimensional bow wave. A supersonic aircraft similarly generates a three-dimensional shock wave. A shock wave is produced by overlapping spheres that form a cone. The conical shock wave generated by a supersonic craft spreads until it reaches the ground. When the shell of compressed air that sweeps behind a supersonic craft reaches listeners on the ground below, the sharp crack they hear is described as a sonic boom. We don’t hear a sonic boom from slower than sound aircrafts because the sound wave crests reach our ears one at a time.

A common misconception is that sonic booms are produced at the moment an aircraft flies through the “sound barrier.” The fact is that a shock wave and its resulting sonic boom are swept continuously behind an aircraft traveling faster than sound. The aircraft that generated this shock wave may have broken the sound barrier hours ago.

Formula Hooke’s Law: whenever a spring is stretched from its equilibrium position and released, it will move back and forth on either side of the equilibrium position. The force that pulls it back and attempts to restore the spring to equilibrium is called the restoring force. Restoring force (F)= force constant (k) x displacement from equilibrium (x) F= kx

Formula Period of a mass on a spring in simple harmonic motion: Period= 2Π √(m/k) m= mass k= force constant

A hummingbird beats its wings up and down with a frequency of 80 Hz A hummingbird beats its wings up and down with a frequency of 80 Hz. What is the period of the hummingbird’s flaps?

In anticipation of her first game, Alesia pulls back the handle of a pinball machine a distance of 5cm. The force constant of the spring in the handle is 200N/m. How much force must Alesia exert?

As Bianca stands on a bathroom scale, whose force constant is 220N/m, the needle on the scale vibrates from side to side. If Bianca has a mass of 180kg, what is the period of vibration of the needle as it comes to rest? If Bianca goes on a diet, how will this change the period of vibration?

Formula The period of a pendulum depends only upon the pendulum’s length. Period= 2Π √(L/g) L= length g= acceleration due to gravity

A tall, thin tree sways back and forth in the breeze with a frequency of 2 Hz. What is the period of the tree?

World-reknowned hypnotist Paulbar the Great swings his watch from a 20cm chain in front of a subject’s eyes. What is the period of the swing of the watch?

A spider swings in the breeze from a silk thread with a period of 0.6s. How long is the spider’s strand of silk?