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...
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...
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...
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...
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...
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
$$I=\frac{2}{3}MR^2$$...
Homework Statement
Find the acceleration of a uniform solid sphere (of mass ##m## and radius ##R##) rolling without slipping down an incline at angle ##\alpha## using the Lagrangian method.
Homework Equations
Euler-Lagrange equation which says, $$\frac{\partial\mathcal{L}}{\partial...
Homework Statement
Consider a half disk (of uniform density) with the flat end lying on the x-axis, symmetric about the y-axis (i.e. being cut into two quarters by the y-axis). Calculate the moments of inertia about each of the axes.
Homework Equations
$$I_{rr}=\sum_{i}m_ir_i^2$$
The Attempt...
Added in the missing absoute value. I think the reason it must be greater than one in the limit is because for any complex number, we may write it as ##re^{i\phi},## with magnitude ##r##. Given then that ##r## is finite, we have that the limit tends to ##\infty## because of the ##n## in the...
Homework Statement
Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent.
Homework Equations
We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent.
The Attempt at a Solution
Applying the ratio test, we find that...
Homework Statement
A bead of mass ##m## slides (without friction) on a wire in the shape, ##y=b\cosh{\frac{x}{b}}.##
Write the Lagrangian for the bead.
Use the Lagrangian method to generate an equation of motion.
For small oscillations, approximate the differential equation neglecting terms...
As a final question. Supposing that ##\rho## is constant, we have that ##\dot{\rho}=\ddot{\rho}=0,## so the Euler-Lagrange equation for ##\rho## reads $$0\frac{mp\dot{\theta}^2-2mgb\rho}{(m+4mb^2\rho^2)}\to\dot{\theta}=\sqrt{2gb}.$$ Does it physically make sense that change in angular velocity...
The Lagrangian (with the m/2 factor added), is, $$\mathcal{L}=\frac{1}{2}m(\dot{\rho}^2+\rho^2\dot{\theta}^2+4b^2\rho^2\dot{\rho}^2)-mgb\rho^2.$$ So, $$\frac{\partial\mathcal{L}}{\partial\rho}=m\rho\dot{\theta}^2+4mb^2\rho\dot{\rho^2}-2mgb\rho,$$ and...
Homework Statement
A particle of mass ##m## moves without slipping inside a bowl generated by the paraboloid of revolution ##z=b\rho^2,## where ##b## is a positive constant. Write the Lagrangian and Euler-Lagrange equation for this system.
Homework Equations...
Should have been ##\omega_0^2.## Also, I'm not solving for ##a##. Rather I should be solving for the ##c## of the particular solution. So from the $$(b^2+\omega_0^2)c=a,$$ we get $$c=\frac{a}{b^2+\omega_0^2}.$$ We verify this is indeed the particular solution. So the most general solution is...
Homework Statement
An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion.
Homework Equations
##F=m\ddot{x}##
The Attempt at a Solution
We have...
I found a neater way to do it I think. We have that, $$v_r=\frac{dr}{dt}.$$ But, $$\frac{dr}{dt}=\frac{dr}{d\phi}\frac{d\phi}{dt},$$ by chain rule. We know that $$v_{\phi}=r\frac{d\phi}{dt}.$$ But we have a function to find ##r(\phi)##. So we can just find $$\frac{d\phi}{dt}$$ and...
Okay. So we know that ##v_{\phi}## is orthogonal to ##r##. So, based on conservation of angular momentum, we can compute ##v_{120,\phi}##. By conservation of angular momentum, we know that, $$mr_0v_0=mr_{120}|v_{120,\phi}|\sin{90}.$$ We note thath ##r_{120}=3.6r_0.## Plugging in gives...
I'm not exactly sure. I know that ##v_{\phi}## and ##v_r## are orthogonal? Or is this only for circular orbits? If they are orthogonal, then I know the direction of ##v_{\phi}## is orthogonal to the position.
Okay, so we have that, $$r(\phi)=\frac{1.8}{1+0.8\cos{\phi}}.$$ We want to find ##\phi## at ##r_0##. So, just plug in ##r_0## and solve for ##\phi##. We have, $$r_0=\frac{1.8r_0}{1+0.8\cos{\phi}}\to1+0.8\cos{\phi}=1.8\to\cos{\phi}=1\to\phi=0,$$ which is very convenient, because as NFuller said...
Homework Statement
A comet orbits the sun. It's position in polar coordinates is given by, $$r(\phi)=\frac{1.8r_0}{1+0.8\cos{\phi}},$$ where ##r_0## is the position at closest approach. Its velocity at this point is given by ##v_0##. Use the concept of angular momentum to find the following...
Okay, so to do that I have to see how each operator affects the basis, right? I'm not sure how that would work with ##S_{1z}## for example. That is, how do I compute ##S_{1z}|1\,0\rangle## for example. I suppose one way would be to decompose ##|1\,0\rangle## as...
Homework Statement
The Hamiltonian of the positronium atom in the ##1S## state in a magnetic field ##B## along the ##z##-axis is to good approximation, $$H=AS_1\cdot S_2+\frac{eB}{mc}(S_{1z}-S_{2z}).$$ Using the coupled representation in which ##S^2=(S_1+S_2)^2##, and ##S_z=S_{1z}+S_{2z}## are...
As solutions go, how does this look.
We have, $$F=ma=m\frac{dv}{dt}=-F_0e^{Kv}.$$ This can be rearranged to, $$e^{-Kv}dv=-\frac{F_0}{m}dt.$$ Then, we can integrate both sides subject to specified initial conditions. We have, $$\int_{v_0}^{v}e^{-Kv'}dv'=\int_{0}^{t}-\frac{F_0}{m}dt'\rightarrow...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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}##...
Alright, so then, $$\frac{\hbar^2}{2m}(G+Q)^2\tilde{u_Q}(G)+\frac{\beta}{2}\tilde{u_Q}(\frac{2\pi}{a})+\frac{\beta}{2}\tilde{u_Q}(-\frac{2\pi}{a})=E\tilde{u_Q}(G)?$$
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...
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...