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  1. V

    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...
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    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...
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    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...
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    Find the apparent weight

    Almost flying up; where is the problem in my analysis of the problem?
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    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...
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    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...
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    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 $$I=\frac{2}{3}MR^2$$...
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    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.
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    Acceleration of a uniform solid sphere rolling down incline

    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...
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    Moment of inertia of a half disk about an axis

    Yes -- and not that I know of.
  11. V

    Moment of inertia of a half disk about an axis

    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...
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    Show that a series is divergent

    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...
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    Show that a series is divergent

    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...
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    Lagrangian for a bead on a wire

    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...
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    Lagrangian for a particle in a bowl with parabolic curvature

    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...
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    Lagrangian for a particle in a bowl with parabolic curvature

    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...
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    Lagrangian for a particle in a bowl with parabolic curvature

    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...
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    Exponentially driven harmonic oscillator

    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...
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    Exponentially driven harmonic oscillator

    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...
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    Use angular momentum to find the velocity (comet orbit)

    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...
  21. V

    Use angular momentum to find the velocity (comet orbit)

    But I don't know what the potential energy is?
  22. V

    Use angular momentum to find the velocity (comet orbit)

    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...
  23. V

    Use angular momentum to find the velocity (comet orbit)

    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.
  24. V

    Use angular momentum to find the velocity (comet orbit)

    It must be conserved? I mean. Should I calculate, ##\rho(\phi)## by differentiating expression for position?
  25. V

    Linear Momentum to Angular Momentum

    Try to calculate the angular momentum of the bullet. Angular momentum and linear momentum are two separate quantities.
  26. V

    Use angular momentum to find the velocity (comet orbit)

    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...
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    Use angular momentum to find the velocity (comet orbit)

    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...
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    Energy eigenvalues of spin Hamiltonian

    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...
  29. V

    Energy eigenvalues of spin Hamiltonian

    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...
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