Multiplying both sides of “eff-equals-em-ey” by a tiny displacement in the (let’s say) x direction, we get:

We know from the equations of motion,

Substituting for we get,

Now we see that work here is just the change in kinetic energy. When we include non-conservative (NC) friction (as true physical systems do) we get some definitions for work:

The Work and Energy Theorem
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• 
• 

Here, U is the potential energy and E_k is the kinetic energy. Using the definition of the potential energy above, we can find the U associated with any force by integrating. The table below shows a few examples:

There are three laws at the heart of Newtonian mechanics:
• An object keeps its state of motion unless acted upon by an external force
• 
• For every force, there is an equal and opposite force

The second of these laws is the omnipotent and ubiquitous “eff-equals-em-ey” which just means that the sum of all vector forces equals the scalar mass times the sum of all vector accelerations.

For absolutely any classical problem, set up a free body diagram (a simple illustration of the magnitude and direction of all the vectors), add up the forces in each dimension and evaluate. Check it out:

example: ice block

Viewed from above, a 10kg block of ice is resting on a sheet of ice. (Forgive the contrived example. The point is, it’s frictionless!) What is the acceleration in the x and y directions?

First, the x component:

Then the y component:

Then the magnitude of the vector a is:

And the angle from the x axis is:

example: frictionless pulley

What is the acceleration of both weights?

For this problem, our (frictionless) pulley only allows forces in the ±y direction. Let’s look at the problem as a free body diagram:

T here is the tension on the rope and g is always the acceleration due to gravity. On the left we have:

And on the right we have:

Solving the above for T, then plugging it into the first equation, we get:

So, the 10kg weight accelerates down (+y) on the left and the 5kg weight accelerates up (+y) on the right, both with the acceleration a.

You may say, “Of course they accelerate at the same rate, jerk.” Or “The heavier one has to pull the lighter one down.” But these are easy things to overlook. Anytime you’re working out a problem, take a step back and try to think intuitively about the (no pun intended) forces at work. “What’s happening in this situation?” is a great question to keep asking yourself.

The thicket of mathematics has many brambles. Don’t get stuck in the equations!

friction

For some more realistic calculations, let’s work friction into the mix. The two types of friction are static and kinetic and the equations of force are as follows:

Here μ_s is the coefficient of static friction, μ_k is the coefficient of kinetic friction and N is the normal (the “equal and opposite force” to the block’s weight).

μ_s is the force a body must overcome to move from a (static) state of rest. μ_k is the force that decelerates a moving (kinetic) body. The inequality of the coefficients represents the “bump” that you feel when you decelerate and stop completely.

example: inclined plane

What is the acceleration of the blocks?

Like always, let’s take each block separately, create a free body diagram to evaluate the x and y contributions and then add it all up.

For the 10kg block, which only moves in the y direction, we have:

For the 3kg block, we have:

And for the 4kg block, we see the true beauty of the free body diagram. We can set our axes in any direction we choose so the equations don’t get mucked up with superfluous sines and cosines. In this case, we’ll rotate the xy-axes by -30º and call the axes x´and y´:

We get values for N₁ and N₂ right off the bat where N is the unit of force called the Newton (not to be confused with N, the normal vector).

Now just add up all the results (”eff-equals-em-ey” is the sum of all forces) and solve for a:

newtonian gravity

The equation of force for Newtonian Gravity is:

Where,

If an orbiting body with mass M has a period τ, the relationship between the period and the radius of two bodies is:

Which brings us to:

Kepler’s Laws:
• Orbits are elliptical with two foci; one being the mass M
• An orbiting body covers equal area in equal time
• 

ex: Bang, zoom, straight to the moon!

At what distance between the Earth and the Moon does a body have no net acceleration due to gravity?

So the free body diagram at the point looks very simple:

Since there is no net acceleration, the forces to the left and right should be equal, giving:

Here, m is the mass of the thing at the point (it could be an asteroid, a taco, whatever) but you don’t have to worry about it since it cancels. Plugging in the values for the mass of the Earth, the mass of the Moon and the distance between their centers, we get:

Using the quadratic formula, we get the positive and negative roots:

The answer must be the negative root since the positive root is larger than R. That means that the position at which an asteroid, taco, whatever would feel no net acceleration from the Earth or the Moon is:

That’s nine-tenths the distance from the Earth to the Moon. POW, right in the kisser!

Problem Set 3 » Newton’s Laws

First, let’s do a super-fast review of dimensional analysis since it will be your best method of spot checking your calculations, not to mention a handy way of guessing answers on obnoxiously long multiple-choice tests.

When solving for a variable in an equation, dimensional analysis will tell you if you’re on the right track or not. For example, an object with initial position, initial velocity and acceleration (in one direction) of x_o, v₀ and a respectively, can be found at a time t by the equation:

Regardless of the equation’s significance, we need the value of n.

Using dimensional analysis, or by doing “normal math” on the units, we can find out very quickly. So,

Since x must be in meters, the seconds must cancel, so n must be 2. For absolutely any equation, just jot down the units and cancel everything you can.

Equations of Motion in Cartesian Coordinates

Kinematics is not actually physics in the truest sense since it only answers the “how” of an arbitrary equation, not the “why” of the system’s behavior. But damnit, it’s useful.

The instantaneous velocity of an object as a function of time is given by an infinitesimal change in position over an infinitesimal period of time or:
v(t) = ^dx/_dt

The instantaneous acceleration is just the derivative of the velocity, or an infinitesimal change in velocity over an infinitesimal period of time:
a(t) = ^dv/_dt

And keep in mind that any variable with ₀ (phonetically “nawt”) means the initial value (t = 0).

Here are the kinematic equations of motion:
• 
• 
• 
• 

Keep in mind that there are only so many well-formed questions you can ask with these equations:

projectile motion

Projectile motion is really just a special case of motion where the acceleration is constant. In the y-direction (+ up and - down), the force is a constant g (= -9.8 m/s² = the force of Earth’s gravity).

Here’s one example, but there are several more in Problem Set 2.

Example: Swish!

What initial velocity is needed to make the shot?

Well, the definition of making the shot is: x (y = 3) = 6
Or, x (t = t_swish) = 6 and y (t = t_swish) = 3

First the x component:

Then the y component:

But we know the value of t_swish from our calculation of the x component.
Just plug it in:

uniform circular motion

For uniform circular motion, it’s much easier to speak in terms of radii and angles instead of x and y components. Using the graph below, we’ll get our new variables.

A particle moving in a circle of radius r will sweep an infinitesimal angle dθ and move a distance s. With constant velocities, we get:

(since sine of a very small angle is approximately equal to the angle)

And we get the centripetal acceleration of an object moving in a circle:

Non-linear coordinate systems

Using these radii and angles, we can construct a set of equations for rotational motion using cylindrical coordinates. And then, we can extrapolate to spherical coordinates!

The kinematic equations of rotational motion:
• 
• 
• 
• 

Spherically speaking, the coordinates become:

Problem Set 2 » Kinematics

This means that each component is equal to the product of the vector’s magnitude and the cosine of the angle between the positive axis and the vector.

So the magnitude of the vector and the angle between the vector and the positive axis are, respectively,

Vector Addition and Subtraction

To add or subtract vectors, just add or subtract the components:

The geometric interpretation is similarly simple. If you start b where a ends, the sum a + b is given by the hypotenuse that makes triangle abc. And if you start a and b at the same point, the hypotenuse gives a — b. Easy!

Vector Multiplication

There are two ways to multiply vectors and each has a nifty geometric interpretation as well.

The dot product of two vectors is a scalar and the rules are the same as vector addition (a • b = a_xb_x + a_yb_y + a_zb_z) but there’s a shortcut!

Since a and b both start at (0, 0, 0), it is easier to calculate the angle θ between them rather than struggle individually with the x, y and z components. We can rotate our axes so that a runs along the x axis and b lies in the xy-plane. This would make the components (a, 0, 0) and (b cos θ, b sin θ, 0). So,

That is, a • b, a scalar, is b times the portion of a that extends in the b direction.

Now’s a good a time as any to introduce the unit coordinate vectors. These are just vectors whose magnitudes are unity and whose directions run along the Cartesian axes.

The unit coordinate vectors are:

(Pronounced “eye-hat,” “jay-hat” and “kay-hat”) This lets us separate the direction of the vector from the magnitude. Now we can express any vector like so:

From the definitions of the unit coordinate vectors, we get the following relations:

The cross product of two vectors is a vector. Written as a matrix, we have:

To put this (or any) matrix into algebraic form, do the following to get your x, y, and z components. For each component in the top row (e.g. i) multiply diagonally to the right (e.g. ia_yb_z) and then to the left (e.g. ia_zb_y) and subtract the second from the first (e.g. ia_yb_z — ia_zb_y). Do this for all the components and evaluate. The above matrix then gives us:

Geometrically speaking, the vector c that results from the cross product of vectors a and b has a magnitude given by the product of |a| and |b| times the sine of the angle from a to b, or |c| = ab sin θ.

The direction is given by the “right hand rule.” If you lay your right hand palm-up along a, then curl your fingers toward b, the resulting vector c is in the direction of your thumb. Take a look:

Properties of the cross product:
• 
• 
• 
• 

From this we get some more unit coordinate vector relations:

Triple Products

The scalar triple product is the dot product of two vectors (since a vector “crossed” with a vector is a vector), which is a scalar:

Now we know that the dot and cross products may be interchanged in the scalar triple product.

The vector triple product comes in handy when evaluating rotating coordinate systems or the rotation of rigid bodies. But more on that later.

It can be shown that a × (b × c) = b(a • c) — c(a • b) but that’s a hell of a lot of LaTeX code. Try it out!

By the way, this is cordially referred to as the “back minus cab” rule for obvious reasons.

Problem Set 1 » Vectors

The Physics GRE consists of 100 five-option multiple-choice questions (some of which are grouped and based on graphical elements or experimental data) and we are given 70 perspiration-drenched minutes to answer as many as possible. According to ETS, the mean score on tests given between July 1, 2000 and June 30, 2003 was a 660 out of 990 or 38.5 out of 100 correctly answered questions.

If you’re already getting antsy, here’s the full practice test (PDF) offered by ETS, but we’ll go through specific problems with explanations, (hopefully) discussion and all that jazz.

The topics covered with corresponding percentages are listed below but you can jump to any category with the links in the left sidebar. As we cover each topic, a permalink will become available so you can jump right to the section.

The test contains:

1. CLASSICAL MECHANICS (20%) such as kinematics, Newton’s laws, work and energy, oscillatory motion, rotational motion about a fixed axis, dynamics of systems of particles, central forces and celestial mechanics, three-dimensional particle dynamics, Lagrangian and Hamiltonian formalism, non-inertial reference frames, elementary topics in fluid dynamics.
2. ELECTROMAGNETISM (18%) such as electrostatics, currents and DC circuits, magnetic fields in free space, Lorentz force, induction, Maxwell’s equations and their applications, electromagnetic waves, AC circuits, magnetic and electric fields in matter.
3. OPTICS AND WAVE PHENOMENA (9%) such as wave properties, superposition, interference, diffraction, geometrical optics, polarization, Doppler effect.
4. THERMODYNAMICS AND STATISTICAL MECHANICS (10%) such as the laws of thermodynamics, thermodynamic processes, equations of state, ideal gases, kinetic theory, ensembles, statistical concepts and calculation of thermodynamic quantities, thermal expansion and heat transfer.
5. QUANTUM MECHANICS (12%) such as fundamental concepts, solutions of the Schrödinger equation (including square wells, harmonic oscillators, and hydrogenic atoms), spin, angular momentum, wave function symmetry, elementary perturbation theory.
6. ATOMIC PHYSICS (10%) such as properties of electrons, Bohr model, energy quantization, atomic structure, atomic spectra, selection rules, black-body radiation, x-rays, atoms in electric and magnetic fields.
7. SPECIAL RELATIVITY (6%) such as introductory concepts, time dilation, length contraction, simultaneity, energy and momentum, four-vectors and Lorentz transformation, velocity addition.
8. LABORATORY METHODS (6%) such as data and error analysis, electronics, instrumentation, radiation detection, counting statistics, interaction of charged particles with matter, lasers and optical interferometers, dimensional analysis, fundamental applications of probability and statistics.
9. SPECIALIZED TOPICS (9%)
   1. Nuclear and Particle physics such as nuclear properties, radioactive decay, fission and fusion, reactions, fundamental properties of elementary particles,
   2. Condensed Matter such as crystal structure, x-ray diffraction, thermal properties, electron theory of metals, semiconductors, superconductors,
   3. Miscellaneous such as astrophysics, mathematical methods, computer applications.

And the math topics covered in one form or another include single and multivariable calculus, coordinate systems (rectangular, cylindrical, and spherical), vector algebra and vector differential operators, Fourier series, partial differential equations, boundary value problems, matrices and determinants, and functions of complex variables.