Recent content by vbrasic

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    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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