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  1. C

    Two particles' spin Hamiltonian

    Homework Statement Hi, I'm trying to familiarize with the bra-ket notation and quantum mechanics. I have to find the hamiltonian's eigenvalues and eigenstates. ##H=(S_{1z}+S_{2z})+S_{1x}S_{2x}## Homework Equations ##S_{z} \vert+\rangle =\hbar/2\vert+\rangle## ##S_{z}\vert-\rangle...
  2. C

    Cylinder carrying volume and surface current, H field

    Homework Statement We have an infinite cylinder that, from radius 0 to a, has a volume current density ##\vec{J(r)}=J_{0}(r/a) \hat{z}## , then from a to 2a, it has a material with uniform linear magnetic permeability ##\mu=(3/2)\mu_0## , and at the surface, it has surface current...
  3. C

    2 True/False Questions -- Integral and convergence

    Homework Statement a) If ##f: [0,1] \rightarrow \mathbb{R}## is continous and ##\int^{b}_{a} f(x)dx = 0## for every interval ##[a,b] \subset [0,1]##, then ##f(x)=0 \forall x \in [0,1]## b) Let ##f: [0,\infty) \rightarrow [0,\infty)## be continous. If ##\int^{\infty}_{0} f(x)dx## converges...
  4. C

    Differentiability of a function -- question on bounding

    Homework Statement I need to see if the function defined as ##f(x,y) = \left\{ \begin{array}{lr} \frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\ 0 & (x,y)=(0,0) \end{array} \right.## is differentiable at (0,0) Homework Equations [/B] A function is differentiable at a...
  5. C

    Hoop rolling on cylinder without slipping

    Hi everybody, I know this problem has been posted before, but it envolved Lagrangian methods which I haven't seen yet. I would appreciate any help. 1. Homework Statement A small hoop is rolling without slipping on a bigger cylinder which is stationary. I need to write Newton's Laws and...
  6. C

    Linear and angular momentum

    Homework Statement Two atoms of equal mass m, that move with the same speed but opposite direction, interact when they're in some region R of space, as in fig.1. After the interaction, one of the atoms moves with velocity ## \vec{V1} ## as in fig.2. a) Are the linear and angular momentum of...
  7. C

    Velocity of Center of Mass

    Hi. There are two masses connected by a massless bar, and from the unstable equilibrium position shown in the figure is slightly inclined so it falls down, being the final state of the system that both masses are in contact with the surface. There is no friction between the floor and m2. The...
  8. C

    Bead on rotating rod

    Homework Statement A particle of mass m is free to slide on a thin rod. The rod rotates in a plane about one end at a constant angular velocity w. Show that the motion is given by r=Ae^(-γt)+Be^(γt), where γ is a constant which you must find and A and B are arbitrary constants. Neglect...
  9. C

    Pebble on wheel (reprised)

    Homework Statement A wheel of radius R rolls along the ground with velocity V. A pebble is carefully released on top of the wheel so that it is instantaneously at rest on the wheel. Show that the pebble will immediately fly off the wheel if V> sqrt(Rg) The Attempt at a Solution Hi...
  10. C

    A rotating disk with two attached masses that slide without friction

    Homework Statement A disk rotates with angular velocity w. Two masses, Ma and Mb, slide without friction in a groove passing through the cnter of the disk. They are connected by a light string of length L, and are initially held in position by a catch, with mass Ma at distance Ra from the...
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