Homework #2

Due in class Wednesday September 9^th.

Problems from book. First few are things you should be able to do at once, they don't need any new material while some of the later ones do. This accounts for the funny order.

1. (Courtesy of Seth.) A ball is kicked into the air with an initial speed of v₀ at and angle θ above the horizontal. If there is no air resistance then the ball follows a parabolic trajectory that you can all easily find. Derive an expression for the angle φ between the ground and the radius vector to the ball. Compare the maximum value of φ with the value of φ when the ball is at the maximum height above the ground. Would the value of φ at maximum height be the same if you repeated the computation for a ball on the moon?
2. 2-6) An application of simple projectile theory.
3. 2-16) Note that when it talks about projecting a particle up a ramp it means sliding it up the ramp. The particle does not become airborne so the problem is a strictly 1-D problem.
4. 2-21) Just work straight from the definitions of torque and angular momentum.
5. The pair of parametric equations
   x = θ - a cos(θ) and y = a sin(θ)
   define a family of curves called Trochoids. (They are basically Spirograph® curves formed by rolling circles on straight lines.) Use Maple to explore the shapes for different values of the parameter a in the range 0.5—2. The relevance of these curves will become apparent after you solve problem 2-22 below.
6. 2-40) Despite its high number this is not really a mechanics problem. It is really an example of the kind of vector differentiation problem from chapter 1. I recommend using Maple to plot the magnitude of a_n to guide your location of the maxima. It turns out to be very hard to persuade Maple to actually find the positions of the maxima in any simple fashion.
7. 2-11) This should be an extension of one of the in-class examples.
8. 2-25) Time to play with energy a little.
9. 2-30) Straight application problem with a little twist.
10. 2-42) Note that the roughness of the surfaces means that the cube will move without slipping. The whole key to this problem is a really large picture of the system before and after tipping through a small angle. The rest is geometry.