Determine the equations describing (2) and (p) for the Hamiltonian given by A = 2 ² 2 + 1/2H (W² 8² + €₂8 + C) 2μ
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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.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?In a Hamiltonian system, what are the conditions for fixed points?
- solve only ( 4 ,5 ,6 )A block of mass m = 240 kg rests against a spring with a spring constant of k = 550 N/m on an inclined plane which makes an angle of θ degrees with the horizontal. Assume the spring has been compressed a distance d from its neutral position. Refer to the figure. (a) Set your coordinates to have the x-axis along the surface of the plane, with up the plane as positive, and the y-axis normal to the plane, with out of the plane as positive. Enter an expression for the normal force, FN, that the plane exerts on the block (in the y-direction) in terms of defined quantities and g. (b) Denoting the coefficient of static friction by μs, write an expression for the sum of the forces in the x-direction just before the block begins to slide up the inclined plane. Use defined quantities and g in your expression. (c) Assuming the plane is frictionless, what will the angle of the plane be, in degrees, if the spring is compressed by gravity a distance 0.1 m? (d) Assuming θ = 45 degrees and the…(a) What does the quadrupole formula (P) = = = (Qij Q³ ³) compute? Reason the answer. (b) A point mass m undergoes a harmonic motion along the z-axis with frequency w and amplitude L, x(t) = y(t) = 0, z(t) = L cos(wt). Show that the only non-vanishing component of the quadrupole moment tensor is = Im L² cos² (wt). (c) Use the quadrupole formula to compute the power radiated by the emission of gravitational waves. (Hint: recall that (cos(t)) = (sin(t)) = 0 and (cos² (t)) = (sin² (t)) = ½½ for a given frequency 2.)