Which of the following is the conserved quantity if the system having Lagrangian ?=12⋅?(?2+?2)−1⋅12⋅?(?2+?2)
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Which of the following is the conserved quantity if the system having Lagrangian ?=12⋅?(?2+?2)−1⋅12⋅?(?2+?2)
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- (a) Let F₁ = x² 2 and F₂ = x x + y ŷ + z 2. Calculate the divergence and curl of F₁ and F₂. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential. (b) Show that the field F3 = yz î + zx ŷ + xy 2 can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.I cant seem to come to (3/2)a . I set up my integral like this: (4/a^3) ∫ r^2 e^[ (-e/a)^2 ] dr is this correct?Two particles, each of mass m, are connected by a light inflexible string of length l. The string passes through a small smooth hole in the centre of a smooth horizontal table, so that one particle is below the table and the other can move on the surface of the table. Take the origin of the (plane) polar coordinates to be the hole, and describe the height of the lower particle by the coordinate z, measured downwards from the table surface. Here, the total force acting on the mass which is on the table is -T r^ (r hat). Why?
- Explain thisThe scalar triple product of three vectors is a • (b x c). Prove that the scalar triple product will not change when you cyclically permute the three vectors. (i.e., prove that a • (b x c) = b • (c x a) = c • (a x b) )The circumference C of a circle is a function of its radius by C(r) = 2xr. Express the radius of a circle as a function of its circumference. Call this function r(C). r(C) = Preview Find r(187). r(187) = Interpret the meaning: O When the radius is 187, the circumference is r(187) O When the circumference is 187, the radius is r(187)
- Need B and C.Consider the length of the simple pendulum is 1 and the mass of the pendulum bob is m . Obtain the equation for the Lagrangian for the simple pendulum when its kinetic energy is m² ø? and the potential energy is mg1(1-cos ø) . 1 (A) - mi g* – mg}(1– cos ø) 2 1 (В) — mi ф - mgl (1-cos ф) 1 (С) — mі ф + mgl(1-сos ) 1 (D) – ml ở – 2mgl (1– cos ø)i need the answer quickly
- Please obtain the same result as in the book.Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.Evaluate the commutator è = [x², Pe** =?