GRE

I think I'm going to shoot for the Physics GRE.

This is the PDF of the practice exam I cover here.

Problem 1

If you don't know what would happen, you should quit and go home. Seriously. Start over with your physics education.

Difficulty: Physics 101, physics for the mathematically disinclined.

Problem 2

The horizontal speed of the object is not important, since air resistance is neglected.

Vertically, the distance traveled is <math>1/2 g t^2</math>. Since it falls for 2 seconds, plugging that in, we get twice g. (If you can't see why, you need to take algebra over again.)

Difficulty: Physics 121, basic Mechanics

Problem 3

<math>P = I V = V^2 / R</math>

I really can't help you remember these equations. There's only resistors, capacitors, and inductors to worry about in electronics.

Difficulty: Physics 122, basic EM

Problem 4

Difficulty: Basic EM

To figure out the force in EM, you have to remove the thing that is being acted upon and calculate the fields. In this case, the magnetic field is found by sticking your thumb in the direction of the current, and curling your fingers.

The force on a charge is the cross product of the velocity of the charge and the magnetic field. In this case, the two vectors are parallel, so the cross product is zero.

Problem 5

This one tests if you remember the De Broglie relation <math>\lambda = h/p</math>.

Difficulty: Basic QM, either in Modern Physics or Intro to QM

Problem 6

If you've memorized the s, p, d, and f, or know the shape of the periodic table, you'd know that this filling n=2 represents the Ne atom, number 10.

Difficulty: Chemistry or intro to QM

Problem 7

Did you remember your thermo?

Difficulty: Thermodynamics and Statistical Mechanics.

Problem 8

Stumped. How are EM waves in a cavity measured in temperature? I can think of blackbody radiation, something about how the energy per unit area something inside the blackbody.

Two thoughts: (1) Is there energy being constantly added when the ice is placed inside? Or does it echo upon itself until it is dissipated? Since they are talking about units of time, I assume that there is a constant influx of energy.

(2) How does energy, or rather, power, relate to heat? I am thinking since the ice is simply melting, we are simply seeing a phase change which means I don't have to worry about work being done on the ice.

Heat is energy, and the amount of ice you can melt with a given amount of energy would be the same. Double the energy, double the amount of ice melted. (Citation needed.)

So really, the question is (1). To double the temperature, do I simply have to double the energy? Or quadruple or whatever?

Aargh.

OK, the Stefan-Boltzmann law says that the radiated power and temperature are related:

<math>P = \text{something something}T^4</math>

Doubling the temperature means you are multiplying the power by <math>2^4 = 16</math>.

Problem 9

This is Newtonian Mechanics from Newton's point of view.

I am fairly certain that the areas have to be equal (I). This is conservation of angular momentum, I believe. Angular momentum = <math>r \times p</math>, and since <math>m</math> isn't changing, then we must conserve <math>r \times v</math>. The area of the triangle is proportional to the cross product. But this is true only for infinitessimally small slices around the ellipse, not large chunks like we have here. Note that it isn't really triangles either, so maybe the rounded bit is what the actual integral would give us.

II is not EXACTLY correct. The star wobbles around one of the focal points. But it is good enough for government work, since the star is much, much more massive than the planet, and the wobble is hardly detectable except by our most precise instruments. I feel like either way, some professor will bark at me. I feel pedantic today, so I declare it is false. (I was wrong. I guess being close is good enough.)

I don't even know how to approach III. What is a semi-major axis? Ugh.

The major axis is the line that goes through both focii from edge to edge. Semi means half, so semi-major is half that.

The period is T. The semi-major axis is a. How can I relate the two? Maybe angular momentum...

The bottom line is I shouldn't guess on this one. It's either a gotcha (the center of mass is at a focus, not the star), or it's basic common sense (of COURSE the star is at the focus.)

Maybe some Chinese student will get this right.

The answer:

Section 16.8 of HRK seems to cover this in detail.

It lays out the Law of Areas which is I. The Law of Periods says that <math>T^2 = \left ( {e\pi^2 \over GM} \right ) a^2</math> which is III. II is the premise of the Law of Orbits which states that the sun is at one of the focii.

Wow. How would I have remembered that?

Problem 10

This is a simple energy equation. The energy stored in a spring when compressed is <math>1/2 k s^2</math> since the force of the spring is <math>k s</math> and you find the energy by adding up all the work done to compress it, <math>\int ks\ ds</math>.

The energy of motion is <math>1/2 mv^2 = 1/2 ks^2</math>. Jumbling it around algebraically, you get <math>s = \sqrt{m v^2 \over k}</math>.

Problem 11

OK, this one got me.

I looked it up, and wikipedia says that <math>E_n = \hbar \omega (n + 1/2)</math>.

I need to study:

(1) What the heck a harmonic oscillator is (2) How to get the Hamiltonian (3) How to solve Schrodinger's equation given a hamiltonian.

It looks like I used to know this.

Problem 12

The Bohr Model is where the electrons orbit a circle around the nucleus. The angular momentum of each orbit is <math>L_n = n \hbar</math>. Angular momentum relates to linear momentum by <math>\mathbf{L_n} = \mathbf{r_n} \times \mathbf{p}</math>, and since these are circles, <math>L_n = r_np</math>. So the linear momentum is simply <math>L_n/r_n = n \hbar \ r_n</math>.

Problem 13

This is testing your ability to read graphs and translate to equations.

We have two points, (2,10) and (200, 100). The only one that gets close is the first one.