Reading:

Read B & O Sections 1.7 -- 1.8; KB 361 lecture notes 6 -- 9.

Problems for Review:

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1. In problem set 1, problem 1 we solved for the motion of a particle with damping force -bv by direct integration techniques. Here we shall solve for its motion using differential equation techniques.
   1. Rewrite Newton's second law as a differential equation for the velocity v(t). Treating this as a differential equation with constant coefficients, find first the most general solution possible for v(t), and then the solution when .
   2. Now write Newton's second law as a differential equation for the position x(t). Treating this as a differential equation with constant coefficients, find first the most general solution possible for x(t), and then the solution when . Confirm (by differentiating) that this answer agrees with your answer for v(t) in part (a).

2. B & O 1-16.
3. Consider a pendulum of length l and a bob of mass m at its end moving through oil with small and decreasing. The massive bob undergoes small oscillations, but the oil retards the bob's motion with a resistive force . The bob is released at t= 0 with and . Find the angular displacement and velocity as functions of time. Sketch .
4. Again consider the pendulum of question 3, but suppose the retarding force is . Again, for bob released at t= 0 with and , solve for and velocity , and sketch .

Problems to Solve:

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1. In problem set 1, problem 3 and problem set 2, problem 2, we considered the dynamics of a hanging cable sliding off a table. Now consider this system from an ODE approach. Treating Newton's second law for the cable as a differential equation with constant coefficients, solve for the cable's displacement x(t) subject to the initial conditions . Check that your answer agrees with problem set 1, problem 3d.
2. In problem set 1, Review problem 2 we solved for the motion of a particle with damping force by direct integration techniques. Here we shall solve for its motion using differential equation techniques.
   1. Rewrite Newton's second law as a linear differential equation for the velocity v(x). HINT: this differential equation must involve x-derivatives only.
   2. Treating this as a differential equation with constant coefficients, find first the most general solution possible for v(x), and then the solution when . Check that your solution agrees with problem set 1, Review problem 2b.

3. Consider the pendulum in oil of Review problem 3 above. Consider the different initial conditions and , which might arise if the pendulum is pushed abruptly or struck with a hammer at t=0. Find the angular displacement and velocity as functions of time. Sketch .
4. Consider the pendulum in oil of Review problem 3, but suppose the retarding force is . For the initial conditions and , solve for and velocity .

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