A particle of mass m moves in a potential of the form V(x)= 2 cx² cx4 where c is a constant. Determine the range of energies for periodic motion, nonperiodic motion with change of direction, and nonperiodic motion without change of direction. What is the period of small oscillation?

Question 6: The dispersion relation of a system is given by w(k) = 2w, sin, where wo is a constant and n is an integer. 1. Calculate the group velocity vg. 2. Calculate the phase velocity Uph.

Problem 2 Consider the block of mass m, connected to a spring of spring constant k and placed on a inclined plane of angle a. Let la be the length of the spring at equili brium, and r be the elongation. The block oscillates and at the same time is rotating around origin 0, in the plane of the inclined, by a variable angular velocity . 1. Calculate the degrees of freedom of the block 2. What is the kinetic energy of the block 3. What is the potential energy of the block 4. Write the Lagrangian function (don't derive the Euler Lagrange equa- tions) k reference plane m o'

Consider a block of mass m attached to a spring with force constant k, as shown in the figure(Figure 1). The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations. Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block. Part A After the block is released from x=A, it will? Part B If the period is doubled, the frequency is? Part C An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10  s. What is its frequency f? Part D If the frequency is 40 Hz, what is the period T ? Part E Which points on the x axis are located a distance A from the equilibrium position? Part F Suppose that the period is T. Which of the following points on the t axis are separated by the time…