Show your work and box your answers. 1. (20 points) A force is described by the function
 =    − + ̂. a. Sketch the force field in the plane z = 0. b. Determine whether the force is conservative. c. Calculate the work the force does on a particle between the origin and point (2, 8, 0) along the path y = x3. 2. (20 points) Find the force for each of the following potential energy functions:  a.  =  b.  =−   + 2    3. (20 points) A particle with charge q exists in a region with a uniform electric field  = . There is no magnetic field. The particle’s initial velocity is  =  ̂ . The initial position is at the origin. a. Write the differential equation of motion using Newton’s second law. Write it in vector form, and then write an equation for each component. b. Find x(t), y(t), and z(t). 4. (20 points) The axis of rotation for a rod of length L is located at the end of the rod which is " defined to be x = 0. The rod’s density is given by   ! = % &# −  # !, where M is the #$ mass of the rod. a. Calculate the moment of inertia. b. If the rod was a physical pendulum, what would its period be? 5. (20 points) A uniform solid cylinder of mass m and radius b has a few turns of light string wound around it. If the end of the string is tied to a beam and the cylinder is allowed to fall, what is the acceleration of the center of the cylinder? 6. (20 points) An ant is on a record player that is rotating clockwise and speeding up. The ant is in the southeast quadrant.  when the record play is viewed from above. a. Draw vectors for ' , ), and -′ b. Use the right hand rule to determine the directions of the transverse force and the centrifugal force. c. Use formal vector calculations to determine the directions of the transverse force and the centrifugal force. d. Now the record player is rotating with a constant angular velocity, but the ant begins walking outward. When the ant is in the southeast quadrant, which direction will the Coriolis force be in? Use both the right hand rule and formal vector calculations. 7. (20 points) A box of mass m is on a spring (spring constant k) and on an inclined plane as shown in the picture below. Ignore friction. a. b. c. d. Write the Lagrangian in terms of x and / Write the Hamiltonian in terms of p and x. Find the differential equation of motion from the Lagrangian. Find the differential equation of motion from the Hamiltonian. 8. (20 points) A ball of mass m is spinning around at the end of a string in space where there is no gravity as shown in the picture below. The string (length l) cannot change length. l a. b. c. d. Write the Lagrangian in terms of 0 and 0/. Write the Hamiltonian in terms of 12 and 0. Find the differential equation of motion from the Lagrangian. Find the differential equation of motion from the Hamiltonian. 9. (20 points) A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. a. b. c. d. Write the Lagrangian in terms of x, y, / , and / . Write the Hamiltonian in terms of x, y, px, and py. Find the differential equations of motion from the Lagrangian. Find the differential equations of motion from the Hamiltonian.
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