Relativity
The two postulates (rules) of special relativity. To earn credit for this problem you need to list both!
1. The laws of physics are the same in all inertial reference frames.
2. The speed of light in a vacuum is constant for all observers.
The units of each element in this propagating wave equation:
y(x,t) = cos(k*x + ω*t + ϕ)
y: meters
x: meters
k: radians/meter
ω: radians/second
t: seconds
ϕ: radians
The equation for the critical angle for TIR. (Hint: start with Snell's law). Double points: what must be true about the relative refractive indices of the materials for TIR to occur?
θ = arcsin(n₂/n₁) where θ is the critical angle.
The λ of an electron moving at 2×10⁶ m/s. (Hint: the mass of an electron is about 9.11×10-31 kg).
About 0.364 nm.
The difference between nuclear fission and nuclear fusion. Double points: which is the LLE trying to achieve right now?
Nuclear fission splits heavy atoms (uranium/plutonium) into lighter ones to release energy, but results in radioactive waste. Nuclear fusion joins light atoms (hydrogen) into heavier ones (helium), releasing more energy without waste, but is currently experimentally more challenging.
A muon travels at 0.99c and has a proper lifetime of 2.2 μs. State what the lifetime is as measured in Earth's frame.
15.6 μs
A wave on a string has the equation:
y(x,t) = 0.02 cos(3x − 12t) meters.
Find the amplitude, wavelength, frequency, and wave speed. Double points: is this wave traveling left or right?
A = 0.02 m
λ = 2π/3 ≈ 2.09 m
f = 12/2π ≈ 1.91 Hz
v = ω/k = 12/3 = 4 m/s
Bonus: the minus sign in (kx − ωt) means the wave travels in the +x direction (right).
A positive (converging) lens with f = 20 cm has an object 40 cm to the left. Find the image distance and magnification.
This is a '4f' system! So, m = -1 and the image is 40 cm to the right of the lens.
The possible electronic states for ground state Neon (Z=10).
[n l ml ms]
(1, 0, 0, +1/2)
(1, 0, 0, -1/2)
(2, 0, 0, +1/2)
(2, 0, 0, -1/2)
(2, 1, 0, +1/2)
(2, 1, 0, -1/2)
(2, 1, -1, +1/2)
(2, 1, -1, -1/2)
(2, 1, +1, +1/2)
(2, 1, +1, -1/2)
The six flavors of quarks.
Up, Down, Top, Bottom, Strange, and Charm.
Rocket A moves at 0.6c relative to Earth. Rocket B moves at 0.8c relative to Earth in the same direction. Find the velocity of B as measured by A. (Hint: use the relativistic velocity addition formula).
Use: u′ = (u − v)/(1 − uv/c²)
u ' = (0.8c − 0.6c)/(1 − 0.48) = 0.2c/0.52 = 0.385c.
A 1 kg mass on a spring is released from rest at x = 0.2 m. At x = 0.1 m, its speed is 3 m/s. Find k and the total mechanical energy of the system.
Remember, the system's total energy is in potential energy at the maximum amplitude position. So:
Using energy conservation: ½mv² + ½kx² = ½kA². Where A is the maximum amplitude of the system.
At x = 0.1 m and v = 3 m/s: ½(1)(9) + ½k(0.01) = ½k(0.04)
Solving we find that k = 300 N/m and the total energy of the system is 6J!
A fish is 1 m below the surface of a pond where the refractive index of the water is ≈ 1.33. State whether the fish appears closer or farther than it actually is person looking straight down. (Hint: draw a picture!)
The fish appears closer.
Think about the double slit experiment with electrons (the Hitachi double slit experiment). State what happens to the interference pattern if you add a detector to see which slit each electron goes through and what this reveals about QM.
The interference pattern goes away if the detector is added and you get two bands that match up with the slits like classical particles. This reveals that act of measurement collapses the electron's wavefunction and observing which path the electron takes forces it to choose one, destroying its superposition that created interference.
State what antimatter is and what happens when a particle meets its antiparticle. Double points: why does this matter for medical imaging?
Antimatter is matter with all quantum numbers reversed. For example, a positron is an 'anti'-electron, same mass but positive charge. When a particle meets its antiparticle they annihilate completely, converting all their mass to energy as gamma ray photons via E = mc².
Bonus: in a PET scan, a radioactive tracer emits positrons that annihilate with nearby electrons, producing pairs of gamma rays in opposite directions. Detectors surrounding the patient identify these photons and reconstruct where the annihilation occurred, mapping metabolic activity in the body.
Einstein's famous equation is E = mc². A single raisin has a mass of about 1 gram. If you could convert it entirely to energy, find roughly how much energy you would get (what is its rest mass energy?). How does this compare to the atomic bomb dropped on Hiroshima (~6×10¹³ J)?
The rest mass energy of the raisin would be 9×1013J, so roughly 1.5 times the energy of the atomic bomb dropped on Hiroshima.
Two tuning forks are struck simultaneously. One is made out of a slightly denser material than the other. A listener nearby hears a slow wavering in volume. State what this wavering is and what causes it.
Because of the tuning fork material, one produces a lower frequency. What the listener is hearing is a beat frequency which is caused by the interference of the two slightly different frequencies produced by the tuning forks as they move in and out of phase with one another.
Two slits are separated by 0.2 mm and illuminated with 650 nm light. The screen is 2 m away. Find how far apart the adjacent bright fringes are. (Hint:you can use the small angle approximation for this)
Starting from the condition for constructive interference for the double slit setup:
d sinθ = mλ
For small angles: d · y/L = mλ
Solving for y of the m and m+1 fringes, we find:
Δy = 6.5 mm
Note that the spacing between fringes does not depend on the order.
An electron is trapped in an infinite square well of width 0.3 nm. Find the ground state energy and the wavelength of the photon emitted when the electron drops from n=2 to n=1 in eV.
ΔE = E₂ − E₁ ≈ 20.1 eV
λ = hc/ΔE = 1240 eV·nm / 20.1 eV ≈ 61.7 nm
(Please write down h in eV for these problems on your cheat sheet!)
Describe what radioactive half life is conceptually and what main assumption must hold for carbon dating to work/be accurate.
Half life is the time it takes for half of a radioactive sample to decay. This time is constant and unaffected by temperature, pressure, or chemistry.
For carbon dating: living organisms continuously absorb carbon maintaining a fixed C-14 to C-12 ratio. When they die, C-14 decays with a 5730 year half life and is not replenished. Measuring the remaining ratio tells you how long ago the organism died. The main assumption is that the atmospheric C-14 to C-12 ratio has been constant over time.
In the film Interstellar, Cooper spends a few hours on Miller's planet while decades pass on Earth. State what physical phenomenon explains this, and what would need to be true about Miller's planet's environment to cause such extreme time dilation. (Hint: Miller's planet orbits very close to the super massive black hole Gargantua and this question is not only about speed)
The phenomenon is a consequence of general relativity. The gravitational field around Miller's planet warps spacetime so severely that time passes much more slowly on the planet's surface relative to a distant observer (Earth). This is distinct from the special relativistic effect we have seen in this course (velocity based time dilation), though both are at play since the planet must be orbiting at enormous speed to avoid falling into the black hole. For the approximately 1 hour = 7 years ratio shown in the film, the planet would need to be incredibly close to Gargantua's event horizon, where the gravitational time dilation factor approaches infinity (this is discussed in The Science of Interstellar!).
The Tacoma Narrows Bridge famously collapsed in 1940 due to wind induced oscillations. Explain using resonance and damping why this happened.
Every structure has a natural resonant frequency determined by its mass, stiffness, and other physical parameters. The wind passing over the Tacoma Narrows Bridge created periodic vortices (alternating swirls of air above and below the deck) at a frequency that matched the bridge's natural frequency. So, each gust of wind added energy the right moment for the oscillations grew larger and larger until the bridge's structure failed and collapsed.
A thin film of oil (n = 1.45) sits on glass (n = 1.6). Find what minimum thickness produces constructive reflection for 560 nm light and how this differs from a soap film (n = 1.33) in air (let's say a soap bubble).
The main difference is the number of phase shifts at each surface.
For oil on glass:
2nt = λ so t = λ/2n = 560/(2 × 1.45) ≈ 193 nm
For soap bubble:
2nt = λ/2 so t = λ/4n = 560/(4 × 1.33) ≈ 105 nm
Rocky walks through a 1 m wide doorway. Using λ = h/mv, estimate whether Rocky (who weighs 100kg) walking at 1 m/s diffracts noticeably through the doorway.
Rocky's deBroglie wavelength is:
λ = h/mv = 6.626×10-34/(100 × 1) ≈ 6.626×10-36 m
This is much smaller than the doorway so Rocky will not diffract!
The point values for each row reference the first four digits (rounded) of some significant physical/mathematical constants you have used in this course (1602, 3000, 5292, 6626, 8854). For this question name all of the constants referenced by the row point amounts in this Jeopardy game. Double points: list the units as well.
1602: Charge of an electron [C] (e)
3000: Speed of light in a vacuum [m/s] (c)
5292: The Bohr radius [m] (a0)
6626: Planck's constant in [J*s] (h)
8854: Permittivity of free space [F/m] (epsilon0)