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

Set 1.  Due Thursday March 19, 2005.  Expires Monday March 14, 2005 at 5 PM.

• a)  T&M 1-17    b) T&M 1-18
• Show by brute force that   A x (B x C) = B (A dot C) - C (A dot B)
• T&M 1-13
• Find 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

• T&M 1-9
• Write down the Lagrangian, and then find Lagrange's equation of motion of a mass M attached to a spring of force constant k which is fixed to an incline of angle alpha. Determine whether the oscillation frequency is affected by the mass being on the incline.
• Write down the Lagrangian and from it determine the acceleration of two masses M1 and M2 connected by a light cord passing over an ideal pulley. Assume M2 > M1. [ This is a simplification of Example 7.8, p. 247.]

• T&M 2-12
• T&M 2-24
• 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.]
• T&M 2-53
• T&M 2-42. They provide the answers to even numbered problems, so the answer is in the book. You must first draw a sketch showing the rectangle tilted by a small angle theta. Then you must work out the potential energy to second order in theta. The equilibrium is stable if one is at a minimum of PE at theta equals zero. Your expression for PE as a function of theta (to second order in small angles should show this is true.

Set 4 Due Monday March 28, 2005. Expires Wednesday March 25, 2005, 5 PM.

• An 1100-kg car has axles 3.0 m apart. The car CM is located 2/3 of the way toward the front of the car, 1 m horizontally from the front axle and 2 m horizontally from the rear axle. The CM is 0.5 m above the ground. Each of the four springs is rated at 48000 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.
• Find the 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 (standing still, it is 1/2 the car's weight) b) Calculate the 'down' 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. 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.

• For the 'tabletop' problem from Set 4, Let the initial thetadot = 10.0 rad/sec, initial r = 0.50 m, initial rdot equals  -0.1 m/s, and initial theta = 0. The masses remain the same, as does the length of the light cord.

• Write down the 'effective potential U_eff of this system, symbolically. From the effective potential, obtain a symbolic equation which will let you find the turning points of the radial motion of mass m.
• Obtain numerical values
• Find the numerical values for r_max and r_min
• Find the numerical value for r at the lowest point in U_eff (must be between rmax and rmin)
• Find the exact frequency of small oscillations in the radial coordinate (roughly around 1 Hz) [This follows what we did when taylor-expanding a potential earlier in the course. It should closely agree with results of a careful simulation.]
• Set up a coupled pendulum using light string or sewing thread and two equal masses. Record all important lengths (see MJM paper from 1978). Record the pendulum frequency and compare experimental values to the theory. Record the time for energy transfer, and compare to the theoretical value in the 1978 paper. Setting this up at home is fine, but you can if you wish set it up in a lab here at school

• T&M problem 12-8. They give the mode frequencies in the back of the book. You must also get the ampltude ratio for each mode, and include this with your solution. (This one takes quite a lot of setting up.)
• Two identical masses m between three springs. The outside springs are attached to walls. One outside spring has twice the spring constant of the other two:      wall    2k   m   k   m   k   wall
• T&M 8-3
• T&M 8-5
• T&M 8-25
• T&M 8-41

• T&M 10-12 (See Figuire 10-6, p. 397)
• T&M 10-18
• T&M 10-19 (very easy)

• a) Calculate, showing all your work the three principal moments of inertia of a solid ellipsoid of mass M defined by (x/a)^2 + (y/b)^2 + z/c)^2.  b) Show that the moments in part a) reduce to the formula for the sphere when a = b = c = R.
• Consider a uniform rectangular plate of sides a and b and mass M. This plate is forced to rotate about an axis through a diagonal of the plate at W radians/sec. a) Calculate the moments of inertia and show they are given by M/12 a^2,  M/12 b^2, and M/12 (a^2+ b^2).  b) Identify the direction of torque needed to rotate the plate. Make a sketch of the plate and show the forces needed to create such a torque. c) Find the magnitude of the torque T in terms of  a, b, M, and W.
• The fractional difference in the Earth's moments of inertia is 1/300, Earth being slightly flattened at the poles. Use this to find the rate at which Earth's angular velocity vector rotates about the Earth's direction of angular momentum
• Treat a water molecule as a mass of 16 u at the origin, a mass u at a distance L along the x-axis from the origin, and a mass u at a distance L from the origin 110 degrees CCW from the x-axis. a) Locate the CM of this molecule. b) Compute the components of the inertial tensor, taking the CM as the origin. That is, calculate the inertia tensor with respect to the CM. c) Diagonalize the inertia tensor by performing a similarity transformation through an angle of 55 degrees. (The inertia tensor will be diagonal along a symmetry axis of the body, along a line through the CM and 55 degrees CCW from the x-axis.
• A football has principal moments of inertia in the ratio of 1.7:1. The ball is poorly thrown (a 'wounded duck') so it wobbles considerably. The nose of the ball makes a 25 degree angle to the angular momentum direction, and the nose of the ball completes one revolution around the angular momenmtum, direction every 0.90 seconds. Find the rate at which the ball rotates about its own axis (psi-dot).