# Search results for query: *

1. ### Hamiltonian for a 1D-spin chain

Homework Statement [/B] A 1D spin chain corresponds to the following figure: Suppose there are ##L## particles on the spin chain and that the ##i##th particle has spin corresponding to ##S=\frac{1}{2}(\sigma_i^x,\sigma_i^y,\sigma_i^z)##, where the ##\sigma##'s correspond to the Pauli spin...
2. ### How do I calculate the potential created by a dipole

Homework Statement I'm given that there is a positive charge of 1 nC at x=0.25 m and a negative charge of -1 nC at x=-0.25 m. I've calculated the potential created at different points along the x-axis by the positive charge and the negative charge using the formula, $$V=\frac{kq}{|r|},$$ where...
3. ### Distance the hoop travels up the incline

Homework Statement A ring (hollow cylinder) of mass 2.61kg, inner radius 6.35cm, and outer radius 7.35cm rolls (without slipping) up an inclined plane that makes an angle of θ=36.0°, as shown in the figure below. At the moment the ring is at position x = 2.19m up the plane, its speed is...
4. ### Find the apparent weight

Almost flying up; where is the problem in my analysis of the problem?
5. ### Find the apparent weight

Homework Statement A car traveling on a straight road at 9.15m/s goes over a hump in the road. The hump may be regarded as an arc of a circle of radius 10.4m. What is the apparent weight of a 665N woman in the car as she rides over the hump? Homework Equations ##F=ma##; ##a=v^2/r## The...
6. ### Find monic generators of the ideals

Homework Statement Let ##T## be the linear operator on ##F^4## represented in the standard basis by $$\begin{bmatrix}c & 0 & 0 & 0 \\ 1 & c & 0 & 0 \\ 0 & 1 & c &0 \\ 0 & 0 & 1 & c \end{bmatrix}.$$ Let ##W## be the null space of ##T-cI##. a) Prove that ##W## is the subspace spanned by...
7. ### Find the velocity when the ball rolls without slipping

Homework Statement A thin spherical shell is sliding with velocity ##v_0## on a table initial until friction eventually causes it to roll without slipping. Find its translational velocity when the it rolls without slipping as a fraction of ##v_0##. Homework Equations $$I=\frac{2}{3}MR^2$$...
8. ### Moment of inertia of a half disk about an axis

Yeah, it wouldn't make a difference. The integral evaluates to the same result either way.

32. ### Find the point when velocity is momentarily zero

So basically $$\frac{F_0K}{m}t+e^{-Kv_0}=1\rightarrow t=\frac{m}{F_0K}(1-e^{-Kv_0}),$$ when velocity goes to 0?
33. ### Find the point when velocity is momentarily zero

Homework Statement [/B] A block travels in the positive x-direction with some velocity ##v_0##. It is subject to a drag force $$F(v)=-F_0e^{Kv},$$ where ##F_0,\,K## are positive constants. Find the point in time when the velocity of the block momentarily goes to zero. Find how far the block has...
34. ### Applying a time-dependent force to a proton

Nevermind, I caught the problem. I messed up on the bounds of my integral. That is, we have that, $$v(t)=\int_{0}^{t}\frac{F_0}{m_p}\sin{(\omega t')}dt'=-\frac{F_0}{m_p \omega}\cos{(\omega t)}\bigg|_{0}^{t}.$$ I mistakenly assumed that ##\cos{0}=0,## which is obviously not the case. Instead, I...
35. ### Applying a time-dependent force to a proton

So, the only issue with this is that the displacement is not unbounded as the question suggests?
36. ### Applying a time-dependent force to a proton

Here is my understanding of the physics behind the problem. The particle is initially at rest at the origin. Then, we start applying this force to the particle. The force imparts some momentum to the particle, and it begins to move. But then, the force oscillates so it imparts the same amount of...
37. ### Applying a time-dependent force to a proton

But the force is oscillating between ##F_0## and ##-F_0##, so it seems to me that acceleration gained by the proton as a result of the force is also lost as the force oscillates with time. So, the velocity should also be oscillating. I'm not able to find where my math goes wrong either subject...
38. ### Applying a time-dependent force to a proton

Yes, sorry; I've edited that into the question. The proton is indeed at rest initially.
39. ### Applying a time-dependent force to a proton

Homework Statement A proton is initially located at the origin of some coordinate system (at rest), when a time-dependent force, $$F(t)=F_0\sin{(\omega t)},$$ is applied to it, where ##F_0## and ##\omega## are constants. a) Find the velocity and displacement of the proton as functions of...
40. ### Is first Brillouin zone the same as Wigner-Seitz cell?

Essentially, my question is, suppose we have a hexagonal lattice, such that the reciprocal lattice vectors are given by, $$A=2\pi\hat{x}+\frac{2\pi}{\sqrt{3}}\hat{y},$$ and $$B=\frac{4\pi}{\sqrt{3}}\hat{y}.$$ The magnitudes of these are, $$|A|=|B|=\frac{4\pi}{\sqrt{3}}.$$ Is the Brillouin zone...
41. ### Is first Brillouin zone the same as Wigner-Seitz cell?

Homework Statement Not a homework question, but I am attempting to understand what exactly the first Brillouin zone is. Homework Equations The Attempt at a Solution From my textbook, what I'm gathering is that one constructs the first Brillouin zone by constructing a "Wigner-Seitz" type cell...
42. ### Help with calculations using molar mass

The dimensional analysis is already done, right here, but I'll restate it in nicer format. ##\frac{[g]}{[g/mol]}=[g]\frac{1}{[g/mol]}=[g][mol/g]=[mol]##. Note that dividing by a fraction is simply multiplication by it's reciprocal.
43. ### What should the null hypothesis be?

That is precisely what I was suggesting... I don't think it should be two-tailed. Sorry for any miscommunication.
44. ### What should the null hypothesis be?

A two-tailed test is of the form ##\mu\neq\mu_0##. That is, you want to test whether or not the mean efficacy of the placebo is greater than or less than the mean efficacy of the placebo. In your case however, you only care if the drug performs better than the placebo (not worse). I recommend...
45. ### What should the null hypothesis be?

Note that these two possibilities do not encompass all possible outcomes. The null hypothesis, ##H_0##, you've proposed is ##E_{drug}=E_{placebo}##, where ##E## is simply the efficacy of the drug/placebo, while your alternative hypothesis, ##H_a## states, ##E_{drug}>E_{placebo}##. What about if...
46. ### Fourier transform of periodic potential in crystal lattice

Thanks for all the help. Expanding that sum over ##G'## is proving very challenging for me because I'm not sure exactly what I'm summing over...
47. ### Fourier transform of periodic potential in crystal lattice

So for ##G=\frac{2n\pi}{a},## it's, $$\frac{\hbar^2}{2m}(\frac{2n\pi}{a}+Q)^2\tilde{u_Q}(\frac{2n\pi}{a})+\tilde{V}(\frac{2n\pi}{a}-\frac{2\pi}{a})\tilde{u_Q}(\frac{2\pi}{a})+\tilde{V}(\frac{2n\pi}{a}+\frac{2\pi}{a})\tilde{u_Q}(-\frac{2\pi}{a})=E\tilde{u_Q}(G),$$ and for ##G=-\frac{2n\pi}{a}##...
48. ### Fourier transform of periodic potential in crystal lattice

Alright, so then, $$\frac{\hbar^2}{2m}(G+Q)^2\tilde{u_Q}(G)+\frac{\beta}{2}\tilde{u_Q}(\frac{2\pi}{a})+\frac{\beta}{2}\tilde{u_Q}(-\frac{2\pi}{a})=E\tilde{u_Q}(G)?$$
49. ### Fourier transform of periodic potential in crystal lattice

So we have, $$\frac{\hbar^2}{2m}(G+Q)^2+\tilde{V}(G-\frac{2\pi}{a})\tilde{u_Q}(\frac{2\pi}{a})+\tilde{V}(G+\frac{2\pi}{a})\tilde{u_Q}(-\frac{2\pi}{a})=E\tilde{u_Q}(G).$$ We can subsitute ##G=\frac{2n\pi}{a}## into get...
50. ### Fourier transform of periodic potential in crystal lattice

For the second part, we are given $$\psi(x)=\frac{1}{\sqrt{L}}\Sigma_G\tilde{u_Q}(G)e^{i(Q+G)x}.$$ This is the same as $$e^{iQx}\frac{1}{\sqrt{L}}\Sigma_G\tilde{u_Q}(G)e^{iGx}.$$ This is exactly, Bloch's theorem, ##\psi(x)=e^{ipx}u(x),## where ##u(x)## is periodic. We can then write the...