Study Video for Physics AP Exam, Part I

This 22 minute video quickly reviews the main equations you might encounter in the first semester of a high school AP physics course. Topics include:

• the big five motion equations
• Newton's three laws
• circular motion
• kinetic and potential energy
• work and power
• momentum and impulse

Friday Science: First Semester Physics in 20 Equations

I like physics. My son's taking high school physics and faces the AP exam at the end of the year. It occurred to me this morning that first semester high school physics really boils down to understanding the following 20 equations/concepts:

1. Know the basic units (lengths, time, mass) and their decimal forms (kilo-, centi-, milli-).

2. Know how to cancel out labels using multiplication, division, etc.

3. Know soh-cah-toa for vectors
(sin a = opposite/hypotenuse; cos a = adjacent/hypotenuse; tan = opposite/adjacent)
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The Big Five Motion Equations
4. d = vt

5. v₂ = v₁ + at 

6. d = v₁t + 1/2 at²

7. d = (v₁ + v₂)/2 * t

8. v₂² = v₁² + 2ad

(Substitute g for a and you have free fall equations.)
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Newton's Laws
9. First Law - a body in motion wants to stay in motion (so horizontal motion is constant if there is no friction).

10. Second Law - F = ma

11. Third Law - For every action there's an equal and opposite reaction: m₁a₁ = m₂a₂

12. Conservation of momentum, where p = mv (momentum = mass x velocity)

13. Impulse: F x t = mΔv (= change in momentum)

Friction Forces
14. The formula for the force to get something moving is F_static = mu_static * normal force. F = μn (The normal force is simply the reverse of weight, mg).

15. The formula for the frictional force of something moving is F_kinetic = mu_kinetic * normal force.
F = μn

Energy Equations
16. Kinetic Energy: K = 1/2 mv²

17. Potential Energy: U = mgh

18. There is a conservation of energy. If there is no friction, potential energy converts completely to kinetic and vice versa. If some is lost as friction or heat, the total is still the same.

19. Work done equals the difference between the start and end values of kinetic energy (work energy equation: W = K₂ - K₁

20. W = Fd (work = force x distance)

Friday Gen Eds MS13: Electromagnetism

This is the thirteenth post in the math/science part of my "Gen Eds in a Nutshell" series. The Gen Ed series consists of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm on the home stretch through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.

Thus far in the math/science subjects:

• Math/Science Overview
• Basic Types of Numbers
• The Atom and Quantum Physics
• The Periodic Table
• Molecules and Ions
• Chemical Reactions
• The Basic Tools of Algebra
• Heat and Thermodynamics
• Basic Geometry and Trigonometry
• The Basics of Calculus
• The Physics of Motion
• The Forces of Motion

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1. In the winter, I will often touch my wife's arm before I kiss her goodnight, because I have given her a shocking kiss more than once. The reason is that I have built up a static charge somehow. It could be my shoes rubbing against the carpet. When it is warmer, the moisture in the air is greater and such charges disperse more easily. But when it is colder, there is less moisture and the charge is vented on my unsuspecting wife serving as a conductor.

A conductor is a material through which electrons move easily. An insulator is a material through which electrons do not move easily. The flow of electrons through a conductor is called current, and this channeling of electrons is electricity. Benjamin Franklin famously demonstrated in 1752 that lightning was a form of electricity, using a kite in a thunderstorm.

Current, Voltage, Resistance
2. The flow of current is measured by how many electrons are passing any given point in a certain amount of time (technically, how much charge is passing that point). The amount of electrons is measured in coulombs. [1] One coulomb is the charge equivalent of 6.2 x 10¹⁸ electrons.

One coulomb of charge moving past a given point in a second is called one ampere. It makes sense that as a current moves through things one by one, the same amount of current goes through each thing, if it doesn't get branched off. Let's say there is a circuit that starts with a power source (like a battery) and goes through a wire and several things (e.g., a light bulb, a heater, etc) but it does so one at a time and never branches off until the wire comes back finally to the power source. We call this a series circuit because the current goes "around" (circuit) through all the elements one by one.

In this case, the amount of "amps" going through each element in that circuit will be the same.

3. The moving charge of electrons always emerges from the power source at its "negative" pole. It thus returns into the source at the "positive" side. Electricity thus flows from negative to positive in a circuit. It is pushed out of the source by an "electromotive force" or EMF. We call the amount of work done by that force on each coulomb of charge a volt. Voltage is thus the work done per coulomb in a circuit.

Unlike current, each element in a series circuit absorbs a certain amount of work. So while every element of a circuit will have the same number of amps, each element in a circuit absorbs some of the work from the electrons. The volts used by each element thus add up to the total voltage at the power source in a series circuit. [2]

4. We might say that each element in the circuit thus provides a certain resistance to the voltage. Without any such resistance, we would have a short circuit and the power source might burn up. The amount of current going through an element is proportional to the amount of work done on that element. That is to say, the more work done on the element, the more current going through it.

This relationship is captured in a law called Ohm's law [3] If E is the voltage and I is the current, then E = I x R, where R is the resistance of an individual element in a circuit. The unit of resistance in electricity is called the ohm and it is the amount of resistance afforded by something that only allows one amp of current to flow through when one volt of force is applied.

So since the current is the same throughout a series circuit and the voltage is the total difference in work done between the poles of a power source, the total resistance in a circuit adds up just like the total voltage does. The total resistance of a series circuit is the sum of all the individual resistances.

5. Power is the amount of work done in a certain amount of time. Electrical power is measured in watts. From every day life, we know that a 100 watt light bulb burns brighter than a 60 watt one. Power equals the voltage times the current, or P = E x I (where E is the voltage and I is the current). So the higher the voltage, the higher the power exerted. The higher the current, the higher the power exerted. And of course if both the voltage and the current are higher, the power exerted is all the higher.

6. I mentioned the rules for series circuits above. As series circuit is one in which there is a single flow of the current. It does not branch at any point. Accordingly, the current is the same throughout. The resistance adds up per element to make up the total resistance. Similarly, the voltage drop across each element adds up to the total voltage in the circuit.

In a parallel circuit, the current branches at some point, perhaps at several points. In this case, it is the voltage that is the same across every branch. Meanwhile, it is the current in each branch that adds up to the total. [4]

The current in each branch depends on the resistance in each branch. As far as the total resistance, it goes down when resistance is added to individual branches, meaning that the total amperage will go up. You can find the current or resistance in any branch using some variation of the E = I x R formula.

Finding the total current or resistance in a parallel circuit is more complicated. If you find the reciprocal (one over x) of the resistance in a branch, then add them together, then take the reciprocal of that sum, you will have the total resistance of the circuit. You can then use the voltage and the total resistance to find the total current.

7. I have assumed that the circuits we have been talking about have been direct current (DC). Direct current is the kind of current that might come from a battery. In a battery, a chemical reaction is used to create the electromotive force that pushes current through a circuit. Direct current is current that is constant, constant in its voltage and amperage unless something interferes with the circuit somehow.

However, the current that comes from a wall plate is alternating current (AC). This is current that alternates between a positive and negative voltage (usually 110 volts), usually at about 60 Hertz or 60 back and forth alternations per second. Another way to think of alternating current is that the flow of current alternates directions.

Alternating current is used because it is more easy to manipulate--for example, to change voltages. It is also easier to generate high voltages in AC rather than in DC. Since high voltage is more suitable for transferring electricity over longer distances, modern societies generate high voltages in AC and then distribute that electricity to local communities where transformers then reduce the voltage and distribute it to houses. Individual devices may then convert the AC to DC so that the device can function.

Magnetism and Induction
8. Most of us know what a magnet is. You can use one to pick up metal. You might touch a screw to a magnet and it will temporarily become magnetized. Then it will stay on your screwdriver more easily. A compass works because the earth has a certain magnetic field. The metal on the compass will turn toward the north pole.

In the early 1800s, a relationship was discovered between electricity and magnetism. When current runs through something, a magnetic field is created around it. Similarly, if a magnet is moved around a conductor, it can generate a current. The rotation of a magnetic field generated by electricity is the cornerstone of the electric motor.

9. Here are some basic facts about magnetism. Magnetic lines form closed loops, the smallest loops possible. Magnetic lines pass through everything, although they can be directed. Like magnetic polarity repels; opposites attract.

10. By coiling a wire around a conductor, you can create an electromagnetic field by running current through the wire. The "left hand rule" expresses how this works. If you picture wrapping your left hand in the way that the wire is wrapped around, then your thumb will point toward what will become the north pole of the artificial magnet (magnetic fields go from north pole out in a closed loop only to re-enter the magnet at its south pole).

By the same token, if you point your left thumb in the direction of the current in a wire (current moves from negative to positive), your fingers will curl in the direction the magnetic field created around the wire moves.

11. Inductance is the property of a circuit that opposes any change of current. When current changes, creating a magnetic field, a counter EMF is created that opposes that change. This is Lenz's Law: "The voltage induced in a circuit by changing current always opposes the change causing it." This is a parallel to Newton's third law. When current is decreasing, inductance wants to maintain it. When current is increasing, inductance wants to resist it.

The unit of inductance is the henry. 1 henry is the amount of inductance that yields one volt when the current is changing at the rate of 1 ampere per second. The symbol for inductance is L.

Induction is the actual creation of a voltage, which requires motion. Inductance does not require current flow. Induction does. Induction is the action of inducing a voltage when current is changing in a circuit.

Charges and Capacitance
12. Coulomb's Law is the analogy to Newton's Law of Gravitation and tells us how to calculate the force of a charge on another charge. The equation looks quite the same!

F_e = k(q₁q₂)/r²    [5]

Any one charge (e.g., q₁) creates a field that travels through space. Any other charge then reacts wit that field, while the initial charge is similarly acted on by the other charge. The main difference between gravity and the electromagnetic force is that the force of gravity only accumulates, while charges are positive and negative and thus can cancel each other out.

13. Capacitance refers to the capacity of two separated plates to retain a certain charge, almost like a battery. Such plates often have some sort of insulator in between them, called a dielectric. The unit of capacitance is a farad. 1 farad is the capacity to store one coulomb of charge when one volt of potential exists across a capacitor.

So C (capacitance) = Q (charge)/E (voltage).

Capacitance is directly proportional to plate area. The bigger the area, the bigger the capacity for storing charge. It is inversely proportional to plate spacing. The bigger the gap, the less the capacity for storing charge.

A vacuum in between the two plates suggests the least capacitance for two plates, so when a vaccuum or air is the "dialectric" between two plates, we say the dialectric constant is 1. Other insulators between the two plates increase capacitance. Pure water, for example, increases capacitance by 81 times!

The equations for calculating capacitance in a circuit are exactly the same as the formulas for resistance and inductance.

Maxwell's Equations
The most fundamental principles of electromagnetism were captured by James Clerk Maxwell (1831-79) in the mid-1800s. In his famous four equations, he captured and built on the work of others before him like Michael Faraday (1791-1867) and Andre-Marie Ampere (1775-1836).

1. Gauss' law for electric fields is named after Carl Friedrich Gauss (1777-1855) and basically says that an electric charge produces an electric field passing through any closed surface around it and that the amount of electric flux passing through that closed surface is proportional to the total charge contained within that surface.

In another form, this law suggests that electric charge is repelled by positive charge and attracted to negative charge.

2. Gauss' law for magnetic fields is closely related to the previous law. It amounts to saying that the total magnetic flux passing through any closed surface is zero.

3. Maxwell's third equation is Faraday's Law, which we have described above but not named. Faraday's law states that a magnetic field passing through a surface induces an electromagnetic force at the boundary of that surface, and a changing magnetic field induces an electric current.

4. Maxwell's fourth equation is the Ampere-Maxwell equation, which we have also encountered above but did not name. An electric current generates a magnetic field.

Next Post: Math/Science 14: Genetics and Evolution

I intent to read through material from a couple books before the next post, so it may take some time to get to the next one, but I will post some review material in the meantime.

[1] After Charles Augustin de Coulomb (1736-1806).

[2] The idea that the voltage difference across each element in a series circuit add up to the total voltage is called "Kirchhoff's law" for voltage.

[3] Named after Georg Ohm (1789-1854).

[4] Kirchhoff's current law.

[5] I have introduced k as a constant to show the similarity with Newton's Law of Gravitation. The constant is usually expressed as 1/4πɛ₀, where ɛ₀ is the "constant of permittivity of free space," approximately, 8.85 x 10^-12 farads/meter.

Friday Gen Eds MS12: The Forces of Motion

This is the twelfth post in the math/science part of my "Gen Eds in a Nutshell" series. The Gen Ed series consists of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm over half way through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.

Thus far in the math/science subjects:

• Math/Science Overview
• Basic Types of Numbers
• The Atom and Quantum Physics
• The Periodic Table
• Molecules and Ions
• Chemical Reactions
• The Basic Tools of Algebra
• Heat and Thermodynamics
• Basic Geometry and Trigonometry
• The Basics of Calculus
• The Physics of Motion

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Four Fundamental Forces
1. At present, physics talks about four basic forces in nature. The one most familiar to us is gravity, although we will ask in a later post if gravity is really a force or rather the curvature of space around massive objects. Some suggest it is based on an as yet undiscovered particle, the "graviton."

The second force we recognize the most is the electromagnetic force, which is the basis for electricity and magnetism. These are the forces that hold atoms together. These are the forces that cause friction. Ultimately, this force is behind how we move things around or how things react chemically with each other.

The other two forces have to do with the nucleus of atoms and so, while they are absolutely essential to existence, they are completely foreign to our experience. The strong nuclear force is what holds the protons of the nucleus together, even though they have the same charge. The weak nuclear force is what causes a nuclear process by which a neutron decays into a proton. [1]

Newton's Three Laws
2. In the 1600s, Sir Isaac Newton (1643-1727) set forth three "laws" relating to force and motion. We mentioned the first law in the previous entry. A body at rest wants to stay at rest, and a body in motion wants to stay in motion. The reason this does not happen is because of gravity (the things we throw forward fall to the ground and so cannot continue forward) and friction. We mentioned in the previous entry that the reason we fly off a merry-go-round or slide to the side in a car is the fact that our bodies want to stay in motion in the direction they were headed.

3. Newton's second law is often summarized in the formula F = ma. Force equals mass times acceleration. Another way to express the second law is that force is the instantaneous rate of change in the momentum of an object. We know momentum as the fact that our bodies want to stay in motion. If we are running down a hill, we may find it hard to stop because our body has momentum.

Momentum can be expressed by the formula p = mv, momentum equals mass times velocity. Another quantity is known as impulse, which is force multiplied by the amount of time the force is applied.

Another law is the conservation of momentum. The amount of combined momentum before a collision, for example, equals the combined momentum after a collision.

4. Newton's third law is often captured in the statement--"For every action there is an equal and opposite reaction." So when a bug hits the windshield of your car, the force it acts on the windshield equals the force the windshield pushes back on it. But because the car is so much more massive than the bug (F = ma), it experiences a negligible deceleration (negative acceleration). But the bug experiences a massive deceleration with its tiny mass, causing it to go splat.

Although it may be counterintuitive, the floor is pushing back up on us (the "normal" force) to the same extent as our weight is pushing down on the floor. That is why we do not fall through. This might be a good point to mention that weight is a force and is technically different from mass. Mass has to do with how much "stuff" something consists of. Weight, on the other hand, is the force of gravity on that mass.

So our weight will be different on different planets because the force of gravity will be different. This is why we can jump farther distances on the moon. If we substitute g (the acceleration due to gravity) for a in the formula F = ma, we have the formula that our weight = mg.

Newton's Law of Universal Gravitation
5. Newton discovered that every object in the universe exerts a gravitational force on every other object in the universe. In particular, he discovered that this force decreased in relation to the square of the distance between the two objects. And it increased in relation to the two masses times each other. The formula looks like this:

F = Gm₁m₂/r²

The force of gravity between two bodies equals a constant (the number 6.67 x 10^-11) times the product of the two masses, divided by the distance between them squared.

So the reason the earth holds us to the ground is because it is so massive. The moon is less massive, so its gravitational pull on us would be less.

Work, Power, and Kinetic Energy
6. In terms of motion, physics defines work as the distance for which a force is applied, so W = Fd. The standard unit of force is the newton, named for the scientist, so a unit of work might be the "newton-meter," which is also called a "joule" of work.

7. Another distinction often mentioned at this point in your physics journey is the distinction between potential energy and kinetic energy. Potential energy has to do with a situation where work is "waiting" to be done, where energy is on the verge of being set in motion. Let's say I am holding a rock in the air. If I let loose, it will fall. We might say it has a certain potential energy that can easily be put into motion as kinetic energy, the energy of motion. Similarly, if I have a wound up rubber band, it is ready to exchange its potential energy for kinetic energy and work done.

The formula for kinetic energy is 1/2mv².

The relationship between kinetic energy and work can be expressed in what is called the work-energy theorem: the total work done equals the change in kinetic energy at the beginning and end or W_tot = K₂ - K₁ = ∆K. This also embodies another law, the law of conservation of energy. The total energy before and after any event or process will always be the same, even though some of the energy is effectively "lost" as heat (see entry covering the second law of thermodynamics).

8. Power is another category in this discussion, often expressed in watts. One watt is one joule of work being done per second. So power is the change in work done per time or P = W/t . Another way to express this is force times velocity.

Rotation and Force
9. The same basic rules apply to rotational motion as to straight line motion. For example, if we substitute the "angular" distance covered (θ) for the straight line distance and we substitute the "angular" velocity (ω) for the straight line velocity, we find the same distance and velocity equations apply to circular motion as to straight line motion. E.g., θ₂ = θ₁ + ω₁t + 1/2at².

By angular distance, I mean the angle in radians that a point has moved along the circle (in effect, the number of "radiuses" traveled along the circumference of the circle). Angular velocity is usually just the number of radians per second. The formula for the kinetic energy of a rotating body is directly analogous to the formula for straight line motion K = 1/2Iω², where I is called the "moment of inertia." It is basically a way of expressing how the mass of a body functions in relation to it spinning on some axis. There are formulas for calculating what it would be depending on the shape of the body and where the axis is that you are rotating it around

10. Torque refers to a force exerted on something you are causing to rotate, such as when you exert force on a wrench to try to turn a bolt. The torque is the force you exert times the "lever arm," where the lever arm is the distance between the point of rotation and the "line of action" where you are exerting the force.

Summary of Formulas

• momentum: p = mv
• impulse: J = Ft
• kinetic energy: KE = 1/2mv²
• work: W = Fd
• power: P = W/t

Next Week: Math/Science 13: Electromagnetism

[1] In the early history of the universe (about 10^-12 seconds after creation), the electromagnetic and weak forces were probably merged in an "electroweak" force. Even earlier than then (10^-34 seconds after creation), the electroweak force was probably combined with the strong force in a grand unified force of some kind. The search for this grand unified force continues.

Friday Gen Eds MS11: The Physics of Motion

This is the eleventh post in the math/science part of my "Gen Eds in a Nutshell" series. The Gen Ed series consists of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm over half way through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.

Thus far in the math/science subjects:

• Math/Science Overview
• Basic Types of Numbers
• The Atom and Quantum Physics
• The Periodic Table
• Molecules and Ions
• Chemical Reactions
• The Basic Tools of Algebra
• Heat and Thermodynamics
• Basic Geometry and Trigonometry
• The Basics of Calculus

______________________________
1. Physics is the study of the "laws" of nature as they relate to matter and energy. We can roughly divide it into 1) mechanics, which has to do with motion and its related forces, 2) thermodynamics, which has to do with heat and its associated dynamics, 3) electromagnetism, and what I might call 4) fundamental physics, such as relativity and quantum mechanics.

Mechanics also roughly divides into two basic types of discussion: 1) the description of motion (kinematics) and 2) the forces of motion (dynamics). This entry has to do with the first--kinematics, the description of motion. The next entry will deal with the forces of motion.

Straight Line Motion
2. If you have been driving for a while, you will know that if you are driving 60 miles/hour for an hour, you will have driven 60 miles. We can put this into a formula: d = vt. You can find the distance you have traveled by multiplying the velocity at which you are traveling by the amount of time you are going at that velocity.

There are some details we might mention. First, you have to use the right units for this equation to work. So if you are using "miles per hour" for the velocity, you have to use hours for the time. Otherwise, you're not ready to multiply the two together. What you're trying to do is cancel out the "per hour" part with the number of hours. So if you had minutes or seconds in one place and hours in the others, it wouldn't cancel right.

A physicist might also want to clarify that there is technically a difference between speed and velocity. When we talk about "speed," we don't care what direction you're moving in. Who knows, you might be driving back and forth between two points for an hour. In physics, when we talk of velocity, we are talking about speed in a specific direction. [1]

We might also make the formula a little more detailed by adding a starting point d₁, the place where you started. Then the formula is d₂ = d₁ + vt .

3. Acceleration is the change in velocity over time. We know it from driving in a car or taking off in a plane. So if speed is distance per time, then acceleration is distance per time per time or distance per time squared. Just as we easily derived an equation for distance in relation to velocity, we can imagine a similar one for velocity in relation to acceleration"

v₂ = v₁ + at

So if you had an acceleration in meters/second squared and you multiplied it by seconds. One of the seconds in the bottom would cancel with the seconds you were multiplying by. And you would be left with meters/second, which is a velocity.

4. If you understood calculus and integration in the previous entry, we can integrate this equation to get another equation for distance traveled when an object is not moving at a constant speed but at a constant acceleration. It turns out to be:

d₂ = d₁ + v₁t + 1/2at²

Another equation finds the distance by multiplying the average velocity by the amount of time. It looks like this:

Finally, if we take this last equation and move the variables around to come up with an equation for t and then substitute this into the second equation above, we end up with the last of the primary motion equations of physics:

5. These are the primary equations for motion in a straight line, such as if you were driving on a straight road, a line in what we might think of as a horizontal direction. A falling body is another kind of "straight-line motion." As it turns out, when a body is relatively close to the surface of the earth, the acceleration due to gravity is constant.

The acceleration of any body near the surface of the earth is 9.8 meters/second squared (or 32 feet/second squared). If we give this constant (meaning a value that doesn't change) the symbol g , we can express the equations above in relation to a body falling in a straight line.

So the distance something will drop in a given amount of time (treating the initial distance and velocity as zero) is

d = 1/2gt²

Implied in this equation is the fact that all objects fall at the same rate of acceleration, no matter how much they weigh. [2]

6. Projectile motion is a motion in two dimensions. If you hit a baseball into the air, there is motion going on in two different directions, driven by two different forces. First, there is a motion in a horizontal direction, an "x" direction. Then there is a motion in the vertical or "y" direction. Gravity is the force at work in the y direction. The force of the bat drove the ball initially both forward and upward.

You can analyze the initial trajectory of the ball in terms of these two components, the x and y components, using trigonometry. So you hit the ball at a certain angle from the horizontal. If you know the velocity at which the ball leaves the bat at this angle, then you can find the "component" of that velocity that is vertical and the component of that velocity that is horizontal.

The "sine" of that angle is the ratio of the opposite side or y component to the hypotenuse or overall velocity. The "cosine" of that angle is the ratio of the adjacent side or x component to the hypotenuse. The bottom line is that if you multiply the sine of the angle by the overall velocity, you will know the initial vertical velocity component for the ball. And if you multiple the cosine of that angle times the initial velocity, you will know the initial horizontal velocity component for the ball.

Now you can analyze these two components of the ball's trajectory separately. For the horizontal component of the ball's trajectory, we have Newton's first law, which we will discuss in the next post: "a body in motion wants to stay in motion." So the ball would continue forward indefinitely if it could (leaving wind resistance out of consideration for the moment).

What makes the ball's time in the air limited is the fact that gravity will pull the ball back down to the ground. By using the velocity equations above, we can figure out first how much time it will take for the ball's upward motion to stop and for it to start falling back down (v₂ = v₁ - gt), remembering that v₂ will be zero. We know v₁, the initial upward velocity. We know g which is -9.8 m/s (negative because it's pulling downwards). So we can solve for t, the time in the air.

We can then find the distance the ball will travel upward by using the equation d₂ = d₁ + v₁t - 1/2gt². We now know t. We know the initial vertical velocity. We know g. d₁ is however far above the ground the bat hit the ball. After we solve for d₂, then we know how high the ball will go before it starts to fall back down.

Circular Motion
A special kind of motion is circular motion. In popular language, we talk about "centrifugal force," which we relate to the way a body wants to fly off of a merry-go-round or the way our bodies crash into the car door if the driver takes two sharp of a curve or the way a rock flies off a sling if you are spinning it around.

Technically, this is not a force but Newton's first law in action--a body in motion wants to stay in motion. Our body wants to continue in the same direction off the merry-go-round or into the car door or off the sling. Therefore, in order to stay on our circular trajectory, we need a force pulling us toward the center of the circle, called a "centripetal" force.

It turns out that a constant acceleration toward the center of a circle is necessary to keep an object moving in constant circular motion. If you spin a yo-yo in a circle, you must constantly apply a centripetal force with your hand. The formula for that constant "radial" acceleration equals v²/r, where r is the radius of the circle and v is the velocity of the object as it would fly tangentially off its circular path if you were to let go.

Periodic Motion
Another kind of motion that has become extremely important for our understanding of electromagnetism and the quantum world is periodic motion. This is a recurring motion such as when a weight is bouncing back and forth on a spring. It is fundamental to clocks of many kinds, such as when a pendulum swings back and forth or the old watches with springs in them. This back and forth motion is called "oscillation."

Some basic terms here are the "frequency" of the back and forth motion (e.g., how many "cycles per second") and the "period" of one back and forth motion (how much time it takes for one cycle). There is also the "amplitude" of the cycle, how big the displacement is from the resting point.

As it turns out, periodic motion can be quantified in terms of circular motion, since the back and forth happens in a cycle, like a point traveling around a circle. The relationship in its simplest form turns out to be:

x = A cos ωt

In this equation, x is the amount of displacement from the rest point at any time t. A is the amplitude or the maximum displacement. "cos" means the cosine of ωt. ω is then something called the "angular frequency." It is 2π times the frequency (cycles per second). [3]

Next Week: Math/Science 12: The Forces of Motion

[1] We say that velocity is a "vector" quantity because it implies motion in a particular direction, while speed is a directionless "scalar" quantity.

[2] Of course not taking into account the fact that something like a feather will experience an air resistance that will make it fall slower than a truck. However, in a vacuum, a feather and a truck would fall at the same acceleration.

[3] The reason for the 2π is to get the frequency into "radian" units, which measure a circle in terms of how many radiuses along the circle you are. The angular frequency ω is thus measured in radians traveled per second.

Friday Science: Classifying Physics

So I've taken a shot at categorizing math and chemistry. Today's physics.

1. What do physics majors take?

• I didn't really find many university catalogs very helpful.
• So there's a first semester introductory course, typically covering Mechanics, then advanced mechanics later
• Then there's a second semester course, perhaps covering Thermodynamics, then advanced thermodynamics later, statistical mechanics
• Then there's a third introductory course, usually Electromagnetism, then more advanced electromagnetism later
• Optics
• Quantum physics, particle physics, and relativity
• Electronics
• Computational physics
• Astrophysics

2. Here are the classifications of Dewey and the Library of Congress:




















So the Library of Congress adds geophysics and meteorology. I suppose fluid mechanics wasn't exactly mentioned above either.

3. Then there are physics textbooks. Here are the units in one I have:

• Mechanics
• Waves
• Thermodynamics
• Electromagnetism
• Optics
• Modern Physics

4. OK, so here's my recombobulation, with the math cross-overs I can think of:

I. Classical Mechanics

• Kinematics - describing motion, including rotational motion
• Dynamics - the forces that cause motion
• The math required to do these well includes basic algebra, trig, differential and integral calculus; there is at least some minimal treatment of vectors, radian measurement, partial differential equations
• wave mechanics (mechanical, fluid, sound)
• add logarithms to the math of sound waves

II. Thermodynamics and Statistical Mechanics

• Great deal of overlap here with those aspects of chemistry relating to changes of states.
• zeroth, first and second laws of thermodynamics
• natural logarithms and e pop up occasionally
• statistical mechanics gets into summation functions and probabilistic functions
• I suppose meteorology and geophysics go here.

III. Electromagnetism and Optics

• Maxwell's four laws and beyond
• Not much new mathematically, surface integrals

IV. Modern Physics

• relativity, special and general
• Relativity required some new math, especially non-Euclidean geometry
• cosmic physics
• quantum mechanics
• Quantum mechanics involves a lot of linear algebra, including matrix algebra. It involves complex analysis, that is, extensive use of complex numbers. There is e.

V. Tools and Equipment

• There is of course the standard and exotic equipment, including everything from particle accelerators to voltmeters to the Hubble Space telescope.
• Computers have increasingly played a role in doing physics, to where computational physics is an important piece of the puzzle

Friday Science: Relativity and the Absolute

1. I finished the next chapter of Brian Greene's, The Fabric of the Cosmos, a few days ago. I was excited because he filled in some things about relativity I had never really read about.

My first two summaries were:

a. Overview
b. Spinning Space Buckets

2. Greene begins with a little flashback to James Clerk Maxwell, whose beautiful equations conquered electromagnetism in the 1800s.

Maxwell's Equations

Maxwell, following Faraday, suggested that electric fields and magnetic fields (which are deeply related) spread out through space. Maxwell was the one who suggested that light itself was an electromagnetic wave that acted in space. In the late 1800s, the theory was that there was an "ether" that light moved through, like the water that ocean waves move through or the air that sound waves move through.

The problem was that there was no evidence of such an ether. More importantly, the speed of light seemed to be the same no matter where it was found--coming from something stationary, coming from something moving. Normally, speeds add up. A person walking 3 mph on a train moving 50 mph is moving 53 mph in relation to the ground. But light on the ground is 3 x 10⁸ and light on the train is 3 x 10⁸ and light from a plane is 3 x 10⁸.

3. This is of course where Einstein comes in in 1905. Light can be the universal speed limit if space and time contract relative to speed. So the train is just a wee bit smaller from the perspective of the ground as it moves, to compensate for the speed of a flashlight shined by someone riding on it. You're not contracted on it, but it contracts relative to the person observing on the ground. A plane contracts relative the person on the ground a smidge more, so that the light shining from its wings also comes out exactly at 3 x 10⁸ mps, no matter who is looking from whatever frame of reference, moving or not.

"The combined speed of any object's motion through space and its motion through time is always precisely equal to the speed of light" (49).

4. There were some new insights for me into some of the more precise contours of Einstein's theory in this chapter. So not everything is relative in Einstein's theory. "Spacetime" as a whole is an absolute reference point. It can be sliced up differently, but it is the same loaf. Time is sliced up differently in some cases. Space is sliced up differently in some cases. But it is the same loaf of spacetime, which he concludes by the end of the chapter is a thing. (I didn't fully understand this last part of the chapter, but I feel like I'm making progress)

At one point of the chapter, Greene talks about how there is a totality to motion through spacetime. If something is more or less not moving in space, then all of its motion is through time. But if it has a velocity, then some of its motion through time is diverted to its motion through space and time moves more slowly. It's a fascinating idea (48).

5. The last part of the chapter turns to the question of acceleration. Einstein's special theory of relativity only applied to objects moving with a constant velocity. His general theory in 1915 turned to the question of gravity and acceleration.

The fundamental insight here was that gravity is really only a body following the contours of spacetime, which is warped by mass. So a planet bends spacetime, and gravity is basically our bodies wanting to follow the path of the warp. The ground stops us. Free fall is thus nothing different from weightlessness.

His field equations were the result:

Friday Science: Spinning Space Buckets

1. Trying to read through Brian Greene's, The Fabric of the Cosmos, to fill in some blanks on the current state of physics. Last Friday I summarized the first chapter. Today it's the second chapter, "The Universe and the Bucket."

2. Relatively short chapter and review today. The chapter is basically about a debate that Isaac Newton and Gottfried Leibnitz had over whether space really existed or was just a concept we invoked to compare objects to each other. The force of Newton's person in the 1600s made it difficult for anyone to win against him.

So Newton's bucket scenario asked why the water in a spinning bucket first is flat and then concave, but that if the twisted rope holding the bucket slowed down and stopped, the water continued to be concave. Newton's answer that space was absolute. The water had a position relative to absolute space. So you could feel changes in acceleration because of the change in relation to absolute space.

3. Leibniz believed that space wasn't anything. Space is just what we call the relation between things. Leibniz pretty much lost the bucket argument in the 1600s, but Ernst Mach revived it in the 1870s. Mach suggested that it's only the matter in the universe, its collective gravity, that would make us feel ourselves spinning in space. There is no absolute space.

So Mach argued that we would not know any difference between spinning or not spinning in a completely empty universe. In a universe half as full, we would only be half as much aware, etc. "You feel acceleration only when you accelerate relative to the average distribution of other material inhabiting the cosmos" (37).

4. This sets us up for Einstein, who apparently showed that Mach and Leibniz were the winners on this one, not Newton.

The Relationship of Physics to Math (Feynman 2)

Yesterday morning I read the first chapter of Richard Feynman's The Character of Physical Law, a series of lectures he gave at Cornell in 1965, fifty years ago. This morning I read the second chapter, "The Relation of Mathematics to Physics." In this chapter you really catch a glimpse of this man's genius, as well as his uncanny ability to explain things.

This chapter is full of insights that, interestingly enough, I have caught these last twenty years or so at IWU, especially being friends with the likes of Keith Drury and Russ Gunsalus. These are insights that were part of the founding of Wesley Seminary, insights that are hard to catch, hard to communicate, hard to convince. When they train you to be a scholar, they do not teach you to think like Feynman, a physicist. They teach you to think like a mathematician.

1. Feynman of course is not disparaging of mathematicians in the least. Indeed, he ends the chapter by apologizing to the layman for the difficulty of mathematics. "If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in" (58), by which he means math. In the chapter, he references people like me, who read book after book, hoping that the next person will finally be able to explain to me what is going on with quantum physics.

But in the end it is just tough. Either you can hack the stuff or you can't. In the words of Euclid to a king, wanting an easier explanation, "There is no royal road to geometry" (58). In the words of the nineteenth century physicist Jeans, "The Great Architect seems to be a mathematician."

This is a powerful statement: "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature... [there are] people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once."

2. Still, Feynman spends most of the chapter indicating that physics is quite distinct from math. He wonders if the mathematicians will prove to be right in the end, that all of nature can be boiled down to certain axioms, certain "building blocks" of knowledge, as it were. He calls this the Greek approach to math, to start with foundational claims and build all other claims from there.

But we are not there yet at all, he indicates. Even in math, he suggests, you can start at different places and get to the same destination. But in physics, things are much more like the Babylonian way of doing math. By his description, the Babylonian way of doing math is that you know a collection of things that are true.

You do not break them down into truth atoms. You see some connections between these various mathematical truths and you intuit when to use one tool and when to use another. There are many analogies in physics, he says, where there are no clear connections between differing rules at all, but they have an uncanny similarity to each other.

3. This is the state of physics right now. It is impossible in physics right now to know what the first principles are--or whether there even are first principles. When Newton was asked what his theory of universal gravitation meant, he indicated that it didn't mean anything. It simply describes how things move.

I have said this many times here and it is in my philosophy book. Science is a collection of very precise myths that express the mystery of the world's operations.

Feynman gave as an illustration three completely independent and apparently unrelated ways to express the phenomenon of movement. The first was Newton's "action at a distance model," where a force is acting on an object at a distance. But there is a second model that looks only on the mass and potential of the object itself. And there is a third model, Euler's principle of least action. This last one was the inspiration for Feynman's claim to fame in quantum physics. Somehow, particles know to take the path of least action.

These three expressions (myths, if you would) are completely unrelated, apparently, but they all explain the motion of a body from one point to another correctly, at least on a macro-scale. "The correct laws of physics seem to be expressible in such a tremendous variety of ways" (55).

4. Practical theology is much more like physics than it is mathematics, in that regard. There are these macro-truths. It isn't always helpful to try to break them down into fundamental truth atoms. As a Biblehead, the game we often play of trying to break down life into Bible atoms (i.e., proof texts) is really embarrassing. It's really just a silly game.

But I digress. There are times when the "physics" of life and morality isn't working right and you need to bring in the "mathematicians." And sometimes the "mathematicians," while exploring the beauty of thought for its own sake, will generate useful tools for life without thinking of its relevance.

Feynman would say that the physicists need the mathematicians at key points, although most of what physicists do, in his own words, is fly by the seat of their pants, to see if something works in the real world. And, of course, Feynman was also very good at math.

Feynman's The Character of Physical Law 1

The second half of June is often the season when administrators take what they can of their remaining vacation days, so I am relaxing a bit these next two weeks (though I am happily and gratefully continuing to work for the Seminary until the end of August).

1. So this morning I picked up an old book of interest to me by Richard Feynman, The Character of Physical Law. I make no promises to finish it, but I read the first chapter this morning.

Feynman was a curious soul. I blogged through the first eight chapters of a biography of him last summer, then stopped when the school year came on heavy. Perhaps I will try to finish working through it. I am cursed with too many interests, too little intellect, and too little time. I wish there was a chamber you could enter where time outside freezes but you can go in and study for a while.

Einstein-Dirac-Feynman-Hawking--that's my current list of the greatest geniuses of twentieth century physics. Feynman gave these lectures in 1965 at Cornell.

2. Chapter 1 is about the law of gravitation, first set down by Newton. I'm not sure what to expect from this book. I'm looking for some insight behind physical law. My hunch is that Feynman will not be able to tell me. My hunch is that he is going to muse about patterns, scientific development, and curious correspondences. But there are no real answers to why the laws of nature work the way they do.

He mentions some of these mysteries in this chapter. So the inverse square of distance seems to have some deep significance, but we don't know what it is (30). It shows up both in the law of universal gravitation and in the formula for electric charge. But the relationship between these two forces is unknown and different on the level of 10 to the 42nd power (a one with 42 zeros after it).

Newton's law of gravitation also had to be modified a little by Einstein, who with the help of David Hilbert set out a general theory of relativity that included gravity. Yet even here, Feynman noted that this theory could not yet account for gravity on the quantum level. This is one of the cutting edges of physics.

Feynman ends this first lecture with three observations.

a. The first is the correspondence between physical reality and mathematics. I've mentioned the inverse square feature.

A fun story in the chapter is when two individuals using only math and the observed movements of Uranus independently asked two different observatories to turn their satellites toward a part of the sky where they expected to find a planet, Neptune. The rationale was the hypothesis that Uranus' orbit was being affected by another large planet.

Feynman's account is different from others. He basically suggests that the British observatory thought it ridiculous that you could find a planet by math, while the German observatory dutifully looked and found. This is probably a better story but perhaps less true.

b. The second observation is that these discoveries are not exact, as in the need for modification by Einstein and still by someone in relation to quantum gravity.

c. The third, though, is the simplicity of Newton's basic equation and fourthly its universal character.

3. In the meantime, the chapter gives a lovely tale of how the law developed from Copernicus to Kepler and Galileo to Newton to Cavendish and several other scientists up till the twentieth century. A reassuring reminder is that it took some four centuries to unfold the content of this chapter and that these scientists worked a lifetime on them. They seem amazingly brilliant after the fact, and no doubt they were.

But they weren't brilliant every day. We tend to miss the trial and error of their probings and the long story of development.

P.S. I forgot to mention the old idea around the time of Galileo that angels pushed the planets around. (Feynman had a flair for the dramatic, so I probably should confirm this) He suggests that this is not entirely wrong, but that they were wrong on the direction in which the angels were pushing. The angels--gravity--were pushing inward toward the sun rather than pushing behind the planet. After all, a body in motion stays in motion all by itself... another mystery whose reasons we know not why.

Teaching math as needed for physics (for nerds only)

I came across a physics textbook a long time ago that was striking to me because it interrupted the flow of the physics instruction with about 7 gray sections on calculus. As a teacher, this approach intrigues me. It could be used in the teaching of any discipline that has prerequisites.

Although I haven't finished or published examples of this approach. I have used something like it to teach biblical languages and the Seminary uses an approah something like this in its integrative approach to the practice of ministry. I believe it is far more effective than the "building block" model if it is done correctly. It is something like problem-based learning.


But I can imagine putting together the beginnings of a science curriculum this way.

1. Instantaneous velocity and acceleration (while discussing motion)

This is a great place to introduce the basics of finding a derivative, and this is how the Marion/Hornyak (MH) book pictured began. Integration can also be introduced as the inverse process in brief. Both would be introduced simply, to be expanded upon later.

2. Vector addition (motion in more than one dimension)
Vector addition begins to come into play as soon as you hit motion in more than one dimension. I find Young and Freedman's (YF) introduction of all things vector in chapter 1 a teaching problem. The cross product in particular is, I think, a difficult concept to introduce at the very beginning. Wouldn't it be more helpful to introduce the various characteristics of vector addition and multiplication as they arise in specific topics? This is also a place to review some basic trig.

3. Summation (forces)
When you get into forces, you could introduce summation notation and extend integral calculus. MH review definite integrals when they get to the application of Newton's laws.

4. Vector dot product (work)
Young and Freedman introduce dot products in chapter 1, but students don't need it until chapter 6. Why not introduce it there? There are more integrals in the treatment of work. So an integrative approach could be solidifying and extending techniques of integration as the student went along.

5. Partial derivatives, nablus (energy)
It is amazing to me that I wasn't introduced to partial derivatives until my third semester of calculus and I don't remember hearing about the nablus/gradient in four semesters. Yet these are relatively easy concepts I could have learned in a first semester. This is frequently the case. Teaching the calculus in order, you don't get to simple and useful concepts until way down the line.

Yet chapter 7 of YF already introduces these fairly straightforward concepts in their treatment of energy.

6. Integration in two and three dimensions
MH have a calculus 5 section on this before a chapter on angular momentum. YF also have an advanced section involving integration in their chapter on angular velocity.

7. Radian measurement (angular velocity)
YF review radian measurement as they begin their chapter on angular velocity.

8. Cross product
It is not until chapter 10 and the treatment of torque that YF ever use the vector cross product. Why introduce it in chapter 1, when most students will have no idea what it means or what it is for?

9. Taylor series
MH review this third semester calculus topic in between their chapters on gravitation and periodic motion.

10. Differentials
I see some differentials by themselves in YF treatment of thermodynamics.

11. Surface integrals
When you get into Gauss' law regarding electric flux

12. The gradient
More partial differentials when you get to electric potential

13. More cross product
When you get into electromagnetic induction

14. Second order differential equations
MH finally review second order differential equations, a fourth semester calculus topic, as they are digging deeper into electromagnetic waves.

I picture, perhaps, two teachers tag teaming over the course of a year or summer intensive. Problems might circle back around to earlier physics topics after new mathematical concepts were introduced. Similarly, methods of application (e.g., in calculus) could be introduced in the process of doing problems.

Just some ideas for a Monday morning...