1. (50%) WeBWorK set # 3.
2. (20%) There is no general analytical solution for the motion of a 3-body gravitational system. However, there exists an analytical solution for the very special case shown in the Figure on the right. In this Figure, three stars are shown, each of mass m, which move with the same speed in the plane of the page along a circle of radius r. The three stars move in a clock-wise direction. Calculate how long it will take for this system to make one complete revolution.
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3. (15%) A hanging copper wire with a diameter of 2 mm has an initial length of 3 m and hangs vertically from the ceiling of your dorm room. When a 5 kg mass is attached to its end, the wire stretches by 0.425 mm; when a 10 kg mass is attached to its end, the wire stretches by 0.85 mm. The density of copper is 9 g/cm3 and one mole has a mass of 63 g. Find the approximate value of the effective spring stiffness of the inter-atomic force. Explain your analysis and any assumptions you may have made.
4. (15%) A small block of mass m is attached to a spring with stiffness k. The rest length of the spring is L. The other end of the spring is fastened to a fixed point on a low-friction table. The block slides on the table in a circular path of radius R > L. How long does it take for the block to go around once?
5. (Optional; 25% extra credit) Download the VPython program threeBodyOrbit.py from the Physics 141 software area that simulates the 3-body star system discussed in Problem # 1. The program assumes that the stars in this system have a mass equal to the mass of our sun, and the radius of the orbit is equal to the radius of the earth around the sun.
a. Using this program, modify the initial conditions of one of the stars (e.g. its momentum or the direction of its momentum; the location where these initial conditions are defined is clearly indicated in the program) and find at least one set of initial conditions for which you observe a long-lasting orbit, much different from the circular orbit studied in problem 1. Report, via email to professor Wolfs, the initial conditions and attach a screen shot of the trajectories of the stars. Note: make sure that you check if the step size is appropriate for the initial conditions (e.g. if you make the step size smaller by a factor of 5, do you get the same results?)
b. Find at least one set of initial conditions for which one of the objects wanders off without returning. Report, via email to professor Wolfs, the initial conditions and attach a screen shot of the trajectories of the stars. Note: make sure that you check if the step size is appropriate for the initial conditions (e.g. if you make the step size smaller by a factor of 5, do you get the same results?)
Last updated on Thursday, September 27, 2007 8:35
