Text: Thornton and Marion, Classical Dynamics of Particles and Systems, 5th Ed.

• a)  T&M 1-17    b) T&M 1-18  (law of sines and law of cosines)
  
• Show by brute force, rectangular coordinates, that   A x (B x C) = B (A dot C) - C (A dot B)
• T&M 1-24
• T&M 1-27
  
• Calculate (not just look up) the great circle distance from Terre Haute to London
• Suggestion: use rectangular coordinates of two vectors and the dot product
• to find the angle between TH and London
• TH   latitude : 39.4 degrees N, longitude 86.16 W.
• You must find the latitude and longitude (!) of London, and look up the Earth's radius

• Spreadsheet on projectile, RK2, vertical motion with linear air drag, from first class meeting. You ust have a scroll bar controlling the time. Mass = 0.15 kg, launch speed is 10 m/s and linear drag force is given by Fdrag = -0.164 v, where Fdrag is in N and v is in m/s. Email your spreadsheet showing the max height with air drag. I used 500 time steps, with a time interval less than 0.01 sec.
  
• Use the idea that df = grad f dot dr  to find expressions for the gradient in both spherical and cylindrical coordinates. [ You must start by correctly writing dr in each coordinate system.]
• Consider an ant at coordinates (1,0,0) m and a square surface in the y-z plane extending from (0,0,0) to (0,1,1) m. a) Find the solid angle subtended by this 1-m square plane at the position of the ant. [Solid angle is the projection of a given area onto a circle of radius 1, centered at the observation point.]  b) Suppose one corner of the square remains at the origin, but it enlarges to be 1000 m on a side in the y-z plane. Find the solid angle subtended by the plane at the position of the ant under this circumstance.
  
• For a mass M attached to a spring of force constant k, with one end of the spring fixed and the mass moving on an incline of angle alpha, write the Lagrangian, and then find Lagrange's equation of motion.  Find the oscillation frequency. Does the frequency depend on the angle of the incline?
• Example 2.9, p. 71,  two masses M1 and M2 connected by a light cord passing over an ideal pulley. Assume M2 > M1. Write the lagrangian and determine the acceleration of the masses. Be specific about the generalized coordinate you have chosen
  

• T&M 2-12
• T&M 2-18. With quadratic air drag, make sure the drag force component changes sign when the velocity component changes sign. Use an air density of 1.3 kg/m^3, as that gives very close to the book's answer. Do part a) and make sure your answer agrees with that of the book. Then change the launch angle to 39 degrees and find the launch speed so the ball just clears the fence. Email me the spreadsheet or Maple code or whatever.
  
• T&M 2-23. You are welcome to use Maple here, but must use a second method as well (Could be RK2). Give the answer and briefly say how you got it.
  
• T&M 2-42. (The answer is in the back of the book.). You must first draw a sketch showing the rectangle tilted by a small angle theta. Then work out the potential energy to second order in theta. The equilibrium is stable if there is a minimum of PE at theta =0. Your expression for PE as a function of theta should show a minimum at theta = 0 when R>b/2, as it says in the back of the book.
• T&M 2-53
  

Set 4 Due Monday March 23, 2012.

• An 1100-kg car has axles 3.0 m apart. The car CM is located 1.3 m horizontally from the front axle and 1.7  m horizontally from the rear axle. The CM is 0.5 m above the ground. Each of the four springs is rated at 10000 N/m. The car's maximum deceleration on dry pavement is -0.8 g.
• Find the amount each spring is compressed when the car is at rest (2 springs per axle).
• Find the static friction coefficient of the car's tires with the dry pavement
• a ) Use torques with respect to the car's CM and find the normal force exerted on the front axle during maximum deceleration (during the deceleration, once the car 'noses down' the angular acceleration of the car body is zero)  b) Calculate the downward tilt angle of the car during maximum deceleration.
• A small mass m = 0.40 kg slides on a smooth tabletop, attached to a large mass M = 1.6 kg via a light cord of length 1.25 m. Mass M hangs from the light cord which passes through a small hole in the tabletop to the small mass m.  Initially the small mass m has initial coordinates of r = 1.0 m theta = 0. Its initial velocities are rdot = +0.10 m/s and thetadot = +0.70 rad/sec.
• Identify two constants of the motion, and calculate their numerical values
• Determine the minimum value of r (the distance of m from the small hole)
• Determine the maximum value of thetadot anywhere in the motion
• Numerically integrate the equations of motion to verify the results of the previous problem. ( Surprise! My preference is Excel using RK2.)  Email me your solution in Excel or Maple or whatever you used. Be specific about showing that the model matches the minimum rdot and maximum thetadot.

• M&T problem 8-6,
  
• 8-20,
  
• 8-31
  
• 8-41
  

• M&T problem 10-8
• 10-9
• 10-11
• 10-12

• Consider two equal masses M at each end of a light rod of length L. The rod is free to oscillate in a vertical plane, pivoted a distance S above the center of the rod. Write out the lagrangian for this system, with the angle theta to the vertical as the generalized coordinate (take theta to be a small angle. Write out the lagrange's equation for this system, and find an expression for the frequency of small-angle rotations
• Repeat the previous problem, and now let the rod have a mass m. Start with the lagrangian and work it out through lagrange's equations. Find the frequency of small-angle oscillations.
  
• a) T&M 11-1,    b) T&M 11-3
• T&M 11-7. F = -kr, so U = 1/2  kr^2. Take x to be the horizontal displacement from the closest point to the force center. Be careful about writing the kinetic energy of the disc. Use the lagrange's equation to obtain an expression for  the oscillation frequency of the disc, in terms of  M, R, k and d.

• Analyze the stability of the 'sleeping' top by writing the lagrangian (p. 455). Then write out  the three equations of the motion. Show that two of the three give constants of the motion. Apply the constants, and a set of boundary conditions (thetadot = phidot = 0. psidot = omega3, theta  = theta1) to evaluate the constants. Then let theta1 -> 0, and apply the thirad of the lagrange equations and finish the analysis for theta a small angle. You should obtain an 'oscillator' equation as the condition for the top to remain 'sleeping'.
  
• Consider the Earth to be a symmetric oblate spheroid, with the largest I greater than the other two by 1 part in 300. The determine the rotation period of the Earth's angular velocity near the North pole.
• A football is a prolate spheroid whose larger moments of inertia are 1.7 times the smaller one. For a given wobbly forward pass, let the nose of the ball be precessing about the angular momentum vector with a period of 1.35 sec, and the angle between the anglar momentuim vector and the nose of the ball be 18 degrees. Determine the pure spin rate of the ball about its symmetry axis (ignoring the laces).
• Construct a coupled oscillator (per the MJM paper) with thread or light string, and two equal masses. Record all lengths, with a clear diagram of which is which. Record the pendulum mode frequency (three periods, minimum) and the time for energy transfer (max to zero.) Compare with the theory.
• T&M problem 12-3