Physics 261: Problem Set #3

1. (K+K 2.11) Naturally you want to use cylindrical coordinates for this one. Resolving F=ma in to it's component equations you'll find two interesting relations from the r-hat and z-hat directions, which let you solve for the T's.
2. (K+K 2.13) This problem is practice in finding constraints. Write down the implications of massless and inextensible (doesn't stretch) strings, and massless and frictionless pulleys. Remember to include coordinates for each subsystem (don't forget the pulleys!) and write down Newton's 2nd law for each first, then consider constraints.
   In general: If a part of the system moves independently, it should be considered a subsystem, with its own coordinates.
3. (K+K 2.17) Part a): Note the discussion on page 93 about the behavior of friction: For bodies not in relative motion, 0 < f < mu N (these are really < or equal). If you calculate f in terms of N, this will give you an inequality that specifies the maximum angle.
   Part b): If the wedge doesn't slide, then xdoubledot = a. (This should follow from a constraint!) Be careful of the sign of f. Note the book made a small typo in the condition between mu and theta here--- the intent was that the angle be too steep, i.e. tan(theta) greater than mu.
   Part c): When bodies are not in relative motion, the direction of friction OPPOSES the motion that would occur in its absence. That is why there is both a minimum and maximum acceleration without sliding.
4. (K+K 2.19/20) The key is here is to identify the constraints. Make sure the coordinates you choose are inertial (e.g. a coordinate attached to M1 would not be inertial.) Don't forget the horizontal force on M3 and its third-law reaction force on M1. And don't forget the force on M1 from the tension (through the pulley).
5. (K+K 2.34) Polar coordinates of course! If the total string length is constant, how is rdot related to V? (This is the constraint needed.) For part a), consider the tangential (theta hat) component of Newton's 2nd law. You will have contributions from both terms in the acceleration. This leads to a separable, first-order differential equation.
6. (K+K 2.26/27) See the discussion starting on page 97 about Simple Harmonic Motion. All you need to show is that the position variable for the body satifies the usual differential equation for SHM.