Fourth Assignment, due Feb. 17 Solutions
This assignment, consisting of problems from Chapter 7, will give you practice in working problems in generalized coordinates, using the Lagrangian method. Chapter 7 starts "Armed with the ideas of the calculus of variations", which you are not; but you have seen a derivation of Lagrange's equations in lecture, and you don't have to read Chapter 6 to be able to do these problems. All you need to do is
Because the important point of this method is that it allows you to choose the variables you like, I feel problems should not prescribe this choice for you. Therefore if you don't like the book's suggested coordinates, you may use others.
- choose "suitable" variables ("generalized coordinates") for the problem
- write T and U in terms of the variables you chose
- define L = T - U (as in Eq 7.18 or 7.30)
- Work out Lagrange's equations (like Eq 7.31) to get expressions for the second time derivative of your variable(s)
- go through any other steps the book's problem or this assignment calls for.
- Text, problem 7.8. In accordance with the above freedom of choice, you may choose X and x from the start, in effect skipping part (a). But if writing T in those coordinates seems complicated, by all means do part (a) first.
- Text, problem 7.16. This could also be done by d(T+U)/dt = 0, but do it via L, for practice.
- Text, problem 7.18. This problem earns its single star because the author assumes -- without saying so -- that m hangs straight down. Make it more interesting by allowing m to swing like a pendulum. It's simplest if you assume that the top of the pendulum is fixed, say by letting the string pass through a small, fixed ring just below the cylinder. For even more challenge, assume no ring and let the momentary axis about which the pendulum rotates go through the point where the string is tangent to the cylinder.
- Text, problem 7.21. Obviously the energy (which is all kinetic) is not conserved for this motion. What energy-like quantity is conserved?
- Text, problem 7.38.
- Maybe a short essay question after you have completed the above problems. Check back Wednesday evening to see whether there is that question at this location.