A particle of mass m is attracted to a force center with the force of magnitude k/ r2. Use plane polar coordinates and find Hamilton’s equations of motion.
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A particle of mass m is attracted to a force center with the force of
magnitude k/ r2. Use plane polar coordinates and find Hamilton’s equations
of motion.
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- Consider the special shape pictured in the diagram below. It is a cylinder, centered on the origin with its axis oriented along z, and it has been partially hollowed to leave two cone-shaped cavities at the top and bottom of the cylinder. The radius of the object is a, its height is 2a, and the solid part of the object (the shaded region that is visible in the rightmost panel of the illustration above, which shows a drawing of the cross-section of the object) has a uniform volume charge density of po. Assume that the object is spinning counter clockwise about its cylinder axis at an angular frequency of w. Which of the following operations is part of the calculation of the magnitude of the current density that is associated with the motion of the rotating object as a function of r (select all that apply)?Find the electric flux crossing the wire frame ABCD of length , width b and whose center is at a distance OP = d from an infinite line of charge with linear charge density λ. Consider that the plane of frame is perpendicular to the line OP (Fig. ). A d Fig. A D b P 8 CCalculate the flux of the given vector field by evaluating the line integral directly alongthe given curve for the below parts:(a) The vector field is ⃗ F = (x − y)⃗i + x⃗j. The curve is the circle x^2 + y^2 = 1in the xy-plane. Use the parameterization x = cos t and y = sin t.(b) The vector field is ⃗ F = (x − 1)⃗i + y⃗j. The curve is a circle of radius 3centered at (1, 1). The parametric form of this circle is⃗r = (1 + 3 cos t)⃗i + (1 + 3 sin t)⃗j, 0 ≤ t ≤ 2π(c) The vector field is ⃗F = x⃗i + y⃗j. The curve is the line segment from thepoint (0, 1) to the point (1, 3).
- Sketch how water curls down a sink, say, in clock-wise rotation. Draw the resulting vector of the curl-operator applied on this water flow.Consider a 3-body system again. This time, let the particles be the Sun, the Earth, and the Moon. (a) Write the relative vector equations of motion for the Moon relative to the Earth. (b) At the very instance where the following right angle triangle geometry is achieved, compute the dominant and perturbing accelerations. Compare the magnitudes of both accelerations, is it reasonable to model the Moon's motion by considering a two-body problem with the Earth? Explain your answer.Compute the flux of the vector field F = 2zk through S, the upper hemisphere of radius 5 centered at the origin, oriented outward. flux =
- Calculate the flux of the vector field F(x, y, z) = 5i + 5j + zk through the closed circular cylinder of radius 4 centered about the z-axis for -6 ≤ z < 6, oriented away from the z-axis. Note: a closed cylinder has a top and a bottom. Flux = SS F.dĀ=For each of the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f (that is, ∇f=F). F(x,y)=(−3siny)i+(10y−3xcosy)j