Chapter 2: Basics

In this chapter various physics concepts and definitions needed for the study of sound, acoustics and musical instruments are presented. If these terms are already familiar to you, you may wish to skim this chapter and skip to the next one on simple harmonic motion.

In science words have specific narrow definitions that sometimes don't correspond to their use in everyday language. These definitions are relatively easy to memorize but surprisingly hard to apply correctly. The only way to really understand these concepts and how to apply them is to work the examples (group work, homework, simulation exercises, etc.) that are part of this book.

It is also important to realize that science is in the business of measuring things. This means that, unlike a math class, any number you see is a measurement of something and has a unit attached to it. So we never have 9.5 but we can have 9.5 centimeters or a frequency of 9.5 hertz (Hz). This book uses the metric system but other sets of units are mentioned. Here is a useful site that converts metric units to other systems: Conversion Calculator.

Key Terms:

Position, displacement, time, velocity, speed, acceleration, mass, weight, density, linear density, force, tension, Newton's three laws, pressure, the ideal gas law, Bernoulli's principle, energy, conservation of energy, power, the first and second laws of thermodynamics.

2a: Motion and Force

The location of an object is called the position and is generally measured in meters (or centimeters or inches or feet, etc.) starting from an agreed upon starting point (called the origin). Sometimes it is convenient to set up a coordinate system with and x-axis, y-axis and z-axis in order to locate something precisely in three dimensional space. We may want to also know how far something is from where it started or from its normal, at rest position. This is the displacement or displacement from equilibrium and is also measured in meters. For example, when a violin string is plucked or bowed, different points on the string are displaced from their equilibrium positions by different amounts initially.

Speed, v, is the rate of change of position and is measured in meters per second, m/s (or ft/sec or miles per hour, mph, etc.). If there is no acceleration, speed is related to distance traveled as rate times time equal distance; vt = d. Velocity is speed with additional information about which direction something is moving. So if you are traveling at 50 mph (no direction given) you have a speed but if you are traveling 50 mph and headed north-east we call that a velocity.

Acceleration is a word that has a specific definition in science but gets confused with speed and velocity in everyday use. Acceleration is a change in velocity over a given time and is measured in meters per second squared, m/s² (or ft/s², etc.). A car going down the road in the same direction might have a constant speed of 50 mph. If neither the speed or direction changes then the acceleration is zero. If the car speeds up, slows down or changes direction then the velocity has changed and there is acceleration. The gas pedal in your car is both an acceleration pedal (if you push it to the floor or ease off the pedal you change your speed and get faster or slower) and also a speed pedal (if you hold it constant you keep a constant speed with no acceleration). Your brake pedal is also an acceleration pedal; it slows you down. Your steering wheel is also an accelerator control because it changes your direction.

For objects near the surface of the earth there is a special acceleration, called the acceleration of gravity. For something falling in the case where air resistance doesn't slow it down too much, the speed changes by 9.8 m/s every second so we say it has an acceleration of 9.8 m/s per 1s or 9.8 m/s² (or 32 ft/s² in the English system). So if a pot gets pushed off an upper story window ledge it starts from rest (zero speed) and will be falling at a speed of 9.8 m/s after one second, 9.8 m/s + 9.8 m/s = 19.6 m/s after two seconds, 9.8 m/s + 9.8 m/s + 9.8 m/s = 29.4 m/s after three seconds, etc. This acceleration of gravity works the same for any object, regardless of size or weight, as long as air resistance is small and you are near the earth's surface. This is what Galileo figured out by dropping different sized objects off the leaning tower of Pisa; objects accelerate (get faster) at the same rate no matter what their size (again, if you can ignore air resistance).

The amount of 'stuff' an object has regardless of where it is (in space, on earth, on the moon), its 'heft' you might say, is called mass and is measured in kilograms, kg (or sometimes grams where 1000 g = 1 kg). The English units of mass are the slug and the stone but we seldom use them.

A force is a push or shove and is measured in newtons , N, which is a kilogram-meter per second squared, kg m/s². The tension in a guitar string is the force pulling at either end of the string (the force pulling each end is the same and balances out so the string doesn't go anywhere). The English unit of force is the pound which is also the unit used to measure weight. This can lead to some confusion because the pound is also sometimes incorrectly used for mass.

Weight is the force that gravity causes an object to exert on the earth. Weight is measured in newtons in the metric system or pounds in the English system. This is how mass and weight get confused; in everyday use we use weight and mass interchangeably which, strictly speaking is incorrect. For example in the grocery store you may see a bag of sugar that says 5 lbs or 2.3 kg which is really comparing different things; lbs measures weight but kg measures mass which is different. What the label should say is "this item weights 5 lbs or 22.3 newtons" (because 2.3 kg times 9.8m/s² is 22.3 newtons which is a unit of weight). Or it could say "this item has a mass of 2.3 kg or 0.16 slug" (5 lbs divided by 32 ft/s² is 0.16 slugs and slugs and kg measure mass). So if you see something that claims to have a weight of 2.3 kg (or 5 lbs) remember what that really means is its mass is 2.3 kg.

2b: Newton's Three Laws

Newton came up with several important laws of physics, three of which we may find useful for the physics of sound.

First Law: Objects continue at rest or in straight-line motion with constant velocity unless an unbalanced force acts on them. You might think that if no forces act the object is stationary. But this is only one case; the first law also says that if all the forces add to zero the object could still be moving at constant velocity. So why do you need to run the motor in your car if you are going down the road in the same direction at the same speed? The motor supplies enough force to overcome friction. Not more force than friction but a force exactly equal to friction so that the total force is exactly zero and you continue moving at a constant velocity. Similarly, a performer playing a wind instrument must continue to supply force on the air in the instrument to overcome friction as the air passes through the instrument. Inertia is the term used for the property of an object that causes it continue in straight line motion unless acted on by an unbalanced force.

Second Law: Forces cause accelerations (forces do not cause velocity but changes in velocity). The second law is also known as F = ma where F is the total force acting on mass m, and a is the acceleration of the mass. Note that as was stated in the first law, a net total force is not needed to maintain a constant velocity; the first law says once something is moving it will keep moving unless a force (for example friction) acts to stop it. When you accelerate your car there is a net force on it. But when you reach cruising speed the net force goes to zero (the motor force exactly cancels resistance forces). In this case (when the net force adds to zero) the object (your car) obeys Newton's 1st law and you continue moving in a straight line at constant velocity.

Third Law: Anytime one object exerts a force on a second object, the second object exerts an equal force (but in the opposite direction) back on the first object. Forces always act in pairs on two different objects; an action force acting on one object and a reaction force acting on the second object. Another way to say this is you cannot touch something without it touching you back, and just as hard as you touched it. This seems simple if you are pushing on the wall; the wall obviously pushes back with the same force. But what about the force on a baseball as you throw it (before release)? Is the force pushing back by the baseball the same as the force you apply? YES! The laws of physics are always true with no exceptions. So how can you throw the baseball if the two forces are equal in magnitude but opposite in direction? The key is to realize the action and reaction forces act on different objects. Your force on the ball makes it go forward until it leaves your hand. The equal reaction force back from the ball acts on your hand (not the ball) so you can feel the ball as you throw it.

Video/audio examples:

Newton's first law: egg drop.
Newton's first law (car crash testing) one, two, three. Watch the movement of the dummy's body and head when the car suddenly stops. Because there is no force acting on the head, it keeps moving, according to Newton's 1st law.
Newton's first and second law: sky diving. If you understand this example you have a good idea of how Newton's laws work.
Newton's first, second and third law applied to a bicycle.
Newton's third law: examples on the space station.

2c: Density and Pressure

The density of something is its mass divided by its volume (m/V) and is measured in kilograms per cubic meter, kg/m³ (or sometimes grams per cubic centimeter; g/cm³). So a kilogram of feathers and a kilogram of iron have the same mass and weigh the same but since the kilogram of feathers takes up more space (larger volume) it is less dense. One other version of density we will use is called the linear density which is the mass per length in kg/m. Bass strings on guitars and in pianos have a larger linear density than the strings used for the higher notes. We will see why later on.

Pressure is defined to be a force acting over an area; P = F/A. There are several units of pressure; we will use the pascal, Pa, which is a newton per meter squared, N/m². Other units are the bar; atmospheres; millimeters of mercury, mmHg; inches of water; torr, etc. A larger force over the same area increases pressure but the same force over a larger area decreases pressure. A dull knife does not apply the same pressure as a sharp knife because the area of contact of the dull blade is larger than the area of contact of the sharp blade. Being stepped on by the heel of a high heeled shoe hurts a lot more than if the same person steps on you with a low heel because the same force (the persons weight) applied with a high heel acts over a smaller area so the pressure is much higher. As we will see, the loudness of a sound wave is related to pressure; high volume sound exerts more pressure on average, and therefore more force on the surface of your eardrum.

For gasses in a closed container, pressure and volume are inversely proportional (pressure increases as volume decreases). Pressure and volume are also each directly proportional to temperature (either pressure or volume or both will increase if the temperature increases). These properties are sometimes summarized as the ideal gas law which can be written as PV = nRT. Here P is pressure, V is volume, T is temperature in kelvin, K. The variable n indicates how much gas there is (in moles where a mole is 6.0 × 10²³ atoms or molecules) and R is a constant equal to 3.14 J/mol K.

Pressure in a liquid or gas is the weight of the liquid pushing downward on an area at a given depth (and is measured in the same units as pressure) or P = ρgh where h is the depth, ρ (Greek letter rho) is the density in kilograms per meter cubed, and g is gravitational acceleration. We sit at the bottom of a sea of air that pushes down on us. This pressure is called atmospheric pressure and it varies a little bit from day today because the air above us is moving and also because of changes in temperature and humidity (and so its density changes). When you use a straw you are decreasing the pressure inside the straw and atmospheric pressure outside the straw pushes the liquid up into the straw. This is why a straw would not work in a vacuum. If you are under water, the water above you pushes down on you in addition to the air above the water which pushes down on the water. Since water is much more dense than air, pressure changes a lot faster as you go deeper under water than it does if you change altitude in the air.

Bernoulli's principle says that if the speed of a fluid (liquid or gas) increases, the internal pressure in the fluid decreases. Take a strip of paper one inch wide and 12 inches long. Hold the short end up to your lips and blow. You'll notice that the strip pulls upward to meet the flowing air. This is because the moving air above has a slightly lower pressure than the stationary air below. A similar effect causes baseballs to change direction (curve balls) and airplane wings to have lift. Some wind instruments and the human voice operate in part because of forces due to the Bernoulli effect, as we shall see.

Video/audio examples:

Pressure and a bed of nails: with a balloon, with a person. What would happen to the pressure if the number of nails is reduced? What would this do to the person lying on them?
Bernoulli's principle: Hair dryer. Why does the ping pong ball stay suspended? Soccer ball. Why does the ball not follow Newton's first law and travel in a straight line? Several Examples. Why do the balloon's come together?

2d: Energy and Power

Energy is the capacity to do work and it is measured in joules; J = kg m²/s². Other common units of energy are the Btu (British thermal unit); the calorie (1 cal = 4.18 J); the food calorie (1 kcal = 1000 cal = 4186.8 J); the kilowatt-hour (1 kWh = 3,600,000 J). Notice that the calories listed for food are actually a thousand scientific calories each.

There are many forms of energy (all measured in joules) and you can convert from one form to another. The capacity of a musical instrument for converting mechanical energy of vibration (a vibrating guitar string for example) into vibrations in the air determines how loud the instrument will sound.

Work is one form of energy. Work is defined scientifically as force times the displacement caused by the force; W = Fd. We only count the part of the force that acts in the same direction as the displacement; so a force acting perpendicular to a displacement does not do work. Likewise, if something doesn't move, no work is done.
Kinetic energy is energy of motion. If we do work on a ball (apply a force over a distance) and then release it the ball will have kinetic energy. Kinetic energy is directly proportional to mass (double the mass of an object at the same speed and you have twice as much energy) and directly proportional to velocity squared (double the speed of an object and you have four times as much kinetic energy); KE = 1/2 mv².
Gravitational potential energy is the energy (work) you can get out of an object due to letting it fall. Or if you do work in lifting a mass against the pull of gravity you store up energy that you can get back by letting it fall. Gravitational potential energy is directly proportional to mass and how high it is: GPE = mgh, where g = 9.8 m/s² is the acceleration of gravity.
You can do work on a spring by either stretching it or compressing it a distance x in which case there is stored spring potential energy. The stiffness of a spring is given by a constant, κ, and the energy stored is SPE = 1/2 κ x².
Electromagnetic radiation is energy carried in the form of electromagnetic waves. Examples of electromagnetic waves are light, radio signals, Wi-Fi signals, blue tooth, cell phone signals, x-rays, gamma rays, microwaves, infrared, ultraviolet, etc. The difference between each kind of electromagnetic wave is the size of the wavelength and the energy it carries. Except for visible light, we cannot detect electromagnetic waves. In general we need some electronic device to detect electromagnetic signals. For example a car radio turns electromagnetic waves from the radio station into audible sound waves.
A chemical reaction occurs when two or more atoms interact by re-arranging where their electrons are located (they may share electrons, donate or borrow electrons, or have other complicated interactions). When this happens energy may be emitted or absorbed in the form of heat and/or electromagnetic waves. A burning candle is an example; the molecules making up the candle are interacting with oxygen and giving off heat (increased random molecular energy) and light (electromagnetic energy). The chemical energy stored in a battery is another example; molecules in the battery can combine in a way to give energy to a flow of electrons in a wire.
As Einstein famously showed, there are certain types of changes in the nucleus of some atoms that give off heat. This nuclear energy comes from the atom changing a very small amount of mass directly into energy via E = mc², where c is the speed of light. In other words, in certain special atoms (called radionuclides) something happens to make the atom either randomly split or give off part of its nucleus. If you could weigh the pieces after the reaction you would find a tiny bit of mass was missing. It is this mass that has been turned into energy via Einstein's famous equation. This is the energy used in nuclear reactors and also the energy source of the sun.

In the above examples there is only one mass or object involved. But we know all matter is made of atoms and chemically bound combination of atoms called molecules that are too small to see, even in a microscope. In a solid these atoms are not stationary but vibrate around an equilibrium position. For liquids and gasses they move relative to each other as you saw in the pressure simulation. In both cases the average kinetic energy is proportional to something we call temperature. Temperature is not a type of energy but is proportional to the internal kinetic energy of the molecules that make up a substance and is measured in fahrenheit, ^oF, celsius, ^oC, or kelvin, K.

In addition to random kinetic energy molecules can bend, vibrate and rotate in both solids, liquids and gasses. If we place an object that has a high temperature (high internal random motion) in contact with an object that has a low temperature (low random internal motion) energy will flow from the high temperature object to the low temperature object (the molecules of each will bump into each other so they eventually have the same average random energy). When this happens we call the energy that moves from the hot object to the cold object heat which is measured in joules. Notice that heat and temperature are not the same thing. Heat is a flow of energy (measured in Joules) and temperature is a number in celsius that is proportional to the internal kinetic energy of the molecules making up a substance.

There is one other term, related to energy, which is how fast energy is being used or delivered. The rate at which energy is used or work done is called power and it is measured in watts, W, and horsepower (1 hp = 746 W). In the US we use watts for electrical power but hp for mechanical power. It would make more sense to either measure light bulbs in hp or cars in watts so that everything had the same units. Power is directly proportional to the amount of energy delivered and inversely proportional to the time it takes to deliver the energy; P = W/t. So accelerating your car up to a certain speed will require the same amount of energy regardless of whether you do it slowly or rapidly. But in order to accelerate faster (reach the same kinetic energy in a shorter time) you need a motor that is more powerful.

2e: Two Energy Laws

One very important concept in the physics of sound (and for all of physics) is the law of conservation of energy: Energy cannot be created or destroyed but it can be transformed from one type of energy to another.

Consider the following energy conversions. You do 10 J of work on the spring of a toy gun so that 10 J of potential energy are stored. When you pull the trigger the spring energy is converted to 10 J of kinetic energy and a ball leaves the gun going upward. The kinetic energy gradually turns into gravitational potential energy as the ball goes upward. Halfway up it has 5 J of kinetic energy and 5 J of potential. At the top there is no kinetic left but you have 10 J of gravitational potential energy. On the way down the potential energy is gradually converted to kinetic energy. Just before the ball hits the ground it has 10 J of kinetic energy. Once it hits the ground the 10 J is converted to sound and heat; the ball is just a tiny bit warmer than it was.

You can also trace the original 10 J back in time before you pushed the spring on the toy gun. The energy for moving your arm came from chemical energy stored in your body. This chemical energy came from the plants and/or animals you ate. That energy can eventually be traced back to the nuclear energy released in the form of electromagnetic waves from the sun. Plants absorbed this light energy to form stored chemical energy for you to eat. Energy was never created or destroyed in any of these conversion steps. The law of conservation of energy is also called the first law of thermodynamics.

There is a second energy law that is also very important but we will not need to know the details of how it works. The second law of thermodynamics says that anytime you change energy from one form to another, some of the energy must (that is the law!) end up in a much less useful form, usually as random thermal energy (heat). When the mechanical energy of bowing a violin is turned into vibrational energy in the strings and then that energy is turned into sound energy in the air, a small amount of the total energy always ends up as heat. The violin and surrounding air will be just a tiny bit warmer. Likewise, amplifiers for radios, stereos and electric guitars give off heat because they are converting electrical energy into sound energy; according to the second law the conversion process has to give off some heat.

As can be seen in the picture above, as the result of the second law, sometimes the usable energy is only a small fraction of the total energy used. Your car motor is very inefficient, losing about 74% of the energy as heat. New technology may improve things some but as long as you are burning gasoline the efficiency of a car motor will remain well below 50% because of the second law of thermodynamics. Many other energy conversion processes are more efficient than process which burn a fuel source. For example, large electric motors and generators can be more than 90% efficient in converting electrical energy to mechanical energy or vice versa. The second law still applies but the consequence isn't as severe in the case of electrical conversions.

Video/audio examples

Energy conservation examples: Bowling ball, real roller coaster, animation of roller coaster.