4. Determine approximately the law of motion of a particle of mass m in the field...
Find the law of motion of a particle mass m and zero energy in one dimension in the field U(x) = -Ax^(4) where A is a positive constant. Given the inital position x0, compute how much time does it take for the particle to escape to infinity if the vector of initial velocity of the particle is pointing away from the origin x=0. Describe the motion when the vector of inital velocity of the particle is pointing toward x=0. 3....
The equations of motion for a particle of mass m and electrical charge q under the influence of a uniform magnetic field B perpendicular to the plane of motion are mx" = qBy' and my" = -qBx'. where x and y are the horizontal Cartesian position coordinates of the particle. Suppose that the particle initially satisfies the conditions Solve the initial value problem and sketch out the trajectory of the particle for t Greaterthanorequalto 0.
3. A particle of mass m, charge q, and inital velocity vo is injected into uniform electric field E, find the position trajectory for the motion of the particle. Only after arriving at the vector equations for the trajectory, find the position of the particle after 5s, when E - 2i - 3j +4k vo- (2i -3j + 4k)1000 1.0 × 10-3 6.02 x 1023 q-1.6 x 10-19 (mass of a single H+)
Problem 4*: (Motion along a spiral) A particle of mass m moves in a gravitational field along the spiral z = k0, r = constant, where k is a constant, and z is the vertical direction. Find the Hamiltonian H(z, p) for the particle motion. Find and solve Hamilton's equations of motion. Show in the limit r = 0, 2 = -g.
A particle of mass m is in a potential energy field described by, V(x, y) = 18kx² +8ky? where k is a positive constant. Initially the particle is resting at the origin (0,0). At time t = 0 the particle receives a kick that imparts to it an initial velocity (vo, 2vo). (a) Find the position of the particle as a function of time, x(t) and y(t). (b) Plot the trajectory for this motion (Lissajous figure) using Vo = 1,...
3. (a) Show that when a particle with mass m and charge q enters a magnetic field having its velocity v perpendicular to the direction of the magnetic field B, it will perform a mv circular path of radius R- qB (b) Using the previous result find an expression for the period T of the circular motion. (c) A charged particle moves into a region of uniform magnetic field, goes through half a circle and then exits that region, as...
1. If a particle moves according to a law of motion S(t)=12-6-7, t 20 Where t is measured in seconds and sin meters, (a) Find the velocity of the particle in terms of t. (b) Find the velocity and the speed at time t=1. (c) When is the particle at rest? (d) When is the particle moving to the right and when is it moving to the left? (e) Find the acceleration of the particle at t. (10pts) 2. Evaluate...
4. A particle of mass m 2 kg moves under the potential energy function U(x.y.z)- (kx + 2 k2y2 +3 k3z3) where k 1N. a. Suppose the particle has speed vo3 m/s when it passes through the origin. What will its speed be if and when it passes through the point (1,1.1)? b. Suppose the particle's speed vo at the origin is not known and that the point (1,1,1) is a turning point of the motion (a point where v0)....
3. The Hamiltonian of a particle of mass m and charge q in a static magnetic field may be written 2 where πί Pi-qAi(x). We shall assume that the magnetic field B is uniform, so that AiEijkBjxk is a suitable choice. (a) Find Hamilton's equation of motion for the particle. (Hint: To simplify the algebra, use the chain rule to write9and similarly for p) 8H UT, 0z,, and similarly for Sp use the chain rule to write oz (b) Show...
Question 4: Sphere in Fields Phenomenon: A positively charged sphere with charge q-2.00 x 10-19 C and mass m- 3.25 x 10-27 kg is traveling in a straight line in the (-) direction. The sphere interacts with an external electric field EExt--,4001 블, and an external magnetic field BExt-1.7n Ext out of page The Big Question we are trying to answer is how fast is the sphere moving? We will answer in steps Ext (a) What is/are the relative magnitude(s)...