P chem ll The kinetic energy T of a particle moving in three dimensions is given...
3. (i) Find the kinetic energy of a particle of mass m with position given by the coordinates (s, u, v), related to the ordinary Cartesian coordinates by y z = 2s + 3 + u = 2u + v = 0+03 (ii) Find the kinetic energy of a particle of mass m whose position is given in cylindrical coordinates = = r cos r sine y (iii) Find the kinetic energy of a particle of mass m with position...
Suppose that a scalar field is constant on a surface As shown in the lectures. there are two methods that one might use to obtain the normal to the surface, and they give the same direction (a) Let r(u, v) be a parametric form for the surface S. Use the vector identity to show that Our ar-λ▽u where λ is a scalar field. [Note: no marks will be awarded for simply stating that a term is zero. If it is...
Mechanics. Need help with c) and d) 1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
the answer is why domain of p is 【0,1】in here And the answer continuous like this , the second question is why the surface integral over the top z=2? And how to get the surface integral over the top of the cylinder and the surface integral over the bottom of the cylinder? over the cylindery20 2. 1) First we compute the volume integral using cylinder coordinates r-ρ cos φ, y ρ sino, 2 = ρ with volume element dV =...
need help Find the length of the curve defined by the parametric equations y3In(t/4)2-1) from t 5 tot- 7 Find the length of parametized curve given by a(t) -0t3 -3t2 + 6t, y(t)1t3 +3t2+ 0t, where t goes from zero to one. Hint: The speed is a quadratic polynomial with integer coefficients. A curve with polar equation 14 7sin θ + 50 cos θ represents a line. Write this line in the given Cartesian form Note: Your answer should be...
# Problem 1 # Suppose a point-mass particle with mass, 'm', moving in a gravitational potential, 'U(r)', where 'r' is the distance from the center of the potential. A positional vector and momentum vector of a particle are vec r' and "vec p', respectively. (\vec means vector symbol.) Q1) An angular momentum vector vec J' is defined as vec J = \vec r x \vec p. Show that \vec J is conserved in such a gravitational potential U(r) which depends...
Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice for integrating over disks. Once we choose a coordinate system we must figure out the area form (dA) for that system. For example, when switching from rectangular to polar coordinates we must change the form of the area element from drdy to rdrd0. To determine that rdrde is the correct formula how the edges of...
Particle in a cylindrically symmetrical potential Let p, o, z be the cylindrical coordinates of a spinless 1. (x = ? coso, y = ? sin ?, p 0, 0 <p < 2?). Assume that the potential en of this particle depends only on , and not on ? and z. Recall that: a. Write, in c ylindrical coordinates, the differential operator associated with the Hamiltonian. Show that H commutes with L, and P. Show fr the wave functions chosen...
Exercise 27.1 Are the following functionals distributions? (a) T(p) Ip(0) (b) T(p)= а, а ЕС. Σ φ(n) (0). (c) T(p) n=0 27.2 The space (IR) of test funct i. One is led naturally to require that test functions he and have bounded support. The space of nitely 9 (R) or simply 9 (recall Definition 15.1.7), est functions y differentiable is denoted by of these functions vanishes outside a bounded interval (which depends on e). (İİ) ф is infinitely differentiable in...
1. This quasi-"walkthrough" problem is great practice in cross-products, vectors, and integration. Consider the current loop shown in Figure 2, with B _Bx, and the loop lying in the x - z plane of the page (y points into the page). We wish to find the net torque on this current loop. we'll do this by integrating in θ, the angle shown (a) We'll start with a little segment dl at point p as shown in the figure. What is...