PHYS 380: STUDY GUIDE FOR PART 3

SYSTEMS OF PARTICLES AND ROTATIONS

Outline:

1. Conservation Laws                                               #24, 25

  1. conservation of linear momentum
  2. conservation of angular momentum
  3. conservation of energy
  4. examples: rockets, conveyor belts, planets
  5. examples: collision problems

2. Two-body problem                                                #26, 27, 28

  1. reduced mass
  2. center of mass coordinates
  3. examples: Rutherford scattering
  4. extension: N-body problem
  5. examples: two coupled harmonic oscillators

3. Rotations: general principles                               #29, 30

  1. rigid bodies: translation, orientation
  2. example: simple pendulum
  3. example: compound pendulum
  4. computation: center of mass
  5. computation: moment of inertia

4. Statics

  1. statics of rigid bodies
  2. statics of structures

5. Equilibrium

  1. flexible strings and cables
  2. solid beams
  3. fluids

Homework:

Problem #24:  A bullet of mass, m, is fired at the bob of a pendulum (mass of bob = M) of length, L, and the bullet becomes embedded in the bob and causes the bob (plus bullet) to swing up to an angle, q, with the vertical.  Express the speed of the bullet in terms of the parameters m, M, L, and q, by using the appropriate conservation principles.

 

Problem #25:  Assume a lunar lander has a rocket engine that have an exhaust speed of 1500 m/s, and that 1/3 of the mass of the lander initially is fuel.  The acceleration due to gravity on the Moon is 1/6 that of earth.  How long can the lunar lander hover above the Moon’s surface before running out of fuel?

 

Problem #26:

  a) Starting from KE = ˝ m1 v1˛ + ˝ m2 v2˛  show that KE = ˝ M V˛ + ˝ m v˛

            where M is the total mass and m is the reduced mass; V is the

            center‑of‑mass velocity and v is the relative velocity.

  b)       Starting from L = m1 (r1 x v1) + m2 (r2 x v2)  show that   L = MR x V + mr x v.

  c)       Starting from p = m1 v1 + m2 v2  show that   p = M V.

Problem #27:  In part 2, the period of orbit, t, for an inverse square law force was expressed by the relation:  t2 = 4p2a3˝m/K˝ where a is the semimajor axis distance and K for gravity is GMm.  (This is a generalization of one of Keplar’s laws.)  This applies as long as m is much smaller than M.  Work out a correction for the case where M = m.

 

Problem #28:  For a pair of identical damped coupled harmonic oscillators (m1=m2, b1=b2, k1=k2), find the two normal modes of vibration. 

HINT:  If k3=0, the solution is easy.  Use this solution to help factor the secular equation.

 

 

Extra Credit Problems:

 

Problem #29:  Find the steady state motion of a propeller with moment of inertia, I, that has an applied torque, ta = to(1 + a*cos(wat) and a frictional torque due to air resistance of tf = -bw where to, a, wa, and b are constants.

 

Problem #30:  The balance wheel of a watch consists of a ring of mass, M, and radius, R, with spokes of negligible mass.  The hairspring exerts a restoring torque, ths = -kq.  Find q(t) if initially q(t=0) = qo and w(t=0) = 0.

 

Problem 5‑15 in book

 

Problem 5‑27, parts a & b.

 

Problem 5‑31

 

Problem 5‑33

     Answer in book may not be correct.

 

Study Questions:

1. Be able to do a problem involving a rocket:

  1. from basic principles, derive the equation;
  2. solve it for a particular case.

2. Be able to set up and solve a two-body collision problem.

3. Be able to solve a Rutherford scattering problem.

4. Be able to explain what is meant by "normal modes".

5. Simple Pendulum:

  1. be able to set it up
    1. exactly;
    2. approximately;
  2. solve it
    1. for exact problem: get to elliptic integral; expand it; get series approximation for period;
    2. for approximation.

6. Compound Pendulum: be able to set it up.

7. Center of Mass and Moment of Inertia:

  1. know the basic definitions in terms of summations and in terms of integrals
  2. be able to use the parallel axis theorem.

8. Outline the method of successive approximations used in a string stretched by a single force.

9. For the equations for pressure as a function of distance:

  1. identify the differential equation;
  2. solve it for an incompressible fluid;
  3. solve it for an ideal gas at constant T.

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