# PHYSICS 303: Classical Mechanics

 Fall 2008 Daniel Finley Tues. & Thurs. 9:30 - 10:50 PM , PandA 184 the Required Problem Session, Physics 451-054, meets Tuesday night from 7 to 9 pm [1 credit hour; CR/NC grade]

### Weekly Problem Sessions:    Questions Posed

 Text: Classical Mechanics , John R. Taylor;

1. 26 August:
• Physical Demonstration of the Chaotic Pendulum
• Demonstration with superball, and question as to the relative magnitude of the total acceleration of the ball just before and just after it hit the floor, assuming complete elasticity of the collision and not forgetting about air resistance
• Physical demonstration with an object on the top of a triangular-shaped (wedge) object, all sliding on an inclined plane. Question concerning the relative size of the normal force on the upper object and its weight.
• Supplemental Mathematical Lecture on Scalars, Vectors, Unit Vectors, Matrices, etc., with Questions and Answers
2. 2 September:
• Problems 1.13, 1.18, and 2.5, from your text.
• Given two vectors, a and b, with an angle θ between them, calculate the following vectors and demonstrate them all on a figure:
ab, A = a[ab], B = b[ab], C = AB.
• Using a choice of origin, the East direction as the x-direction, the North direction as the y-direction, and the positive z-direction as up, the surface of a particular hill is given by the function
0 = f(x,y,z) = z - 2xy + 3x2 + 4y2 +18x - 28y -12 .
Where is the top of the hill, how high is it, and what is the direction of the normal at that point?
• Given a plane inclined to the horizontal at an angle θ, a movable block sliding downward along it, which has a bottom edge parallel to the inclined plane but a top edge that is horizontal, and a small rectangular block on top of it, show all forces on all three objects, assuming the existence of appropriate friction at every interface. There are two plausible cases, both of which could be considered:
• the two blocks move together, i.e., so that the interface between the two is static, and
• the two blocks move separately, i.e., so that there is motion at the interface between the two.
For tonight, let us consider ONLY the first possibility, and determine the normal force and the frictional force on the upper block. Assume the coefficient of friction between the two blocks is μ, while the coefficient of friction between the lower block and the inclined plane is given by ν.
• Consider Problem 1.40, and perhaps 2.6 and 2.7, if one finishes the others too quickly.
3. 9 September:
• Problems 2.10, 2.14, 2.30, 2.41, and 2.42
4. 16 September:
• Problems 2.40, 3.4, 3.22, and 3.29
• If a given particle has mass m and location given by r = b(x hat) + vt(yhat), what are its velocity, its acceleration, and its angular momentum? Please also re-express all these quantities in cylindrical coordinates.
• What is the time to fall from rest through a height h for a baseball moving through air with quadratic resistance?
• A baseball, moving with quadratic resistance, is thrown up with a velocity three times the magnitude of the terminal velocity for this medium and object. What is the time required for it to return to earth?
• Prove that a 2-dimensional trajectory, moving through air with quadratic resistance, can move only a finite distance horizontally, no matter how long it travels.
5. 23 September: First Exam, on Chs. 1 & 2.
6. 30 September:
• Problems 4.9, 4.24, 4.28, 4.31, 4.36
7. 7 October: Second Exam, on Chs. 3 & 4.
8. 14 October:
• A pendulum of mass m is attached to a rigid steel rod, of length L, and is swinging underwater. Because of the water there is a resistive force to its motion with magnitude 2mL(g/L)1/2dθ/dt . Set up the equation of motion and solve it for the case where the pendulum is lifted up to some small, initial angle α and released from rest. Present a graph of a phase diagram for it.
• An electric motor weighing 100 kg is suspended by vertical springs which stretch 10 cm when the motor is attached. If the flywheel on the motor is not properly balanced, for what amount of revolutions per minute (rpm) would resonance be expected?
• A tuning fork with frequency of 440 cycles per second, ie., the note A above middle C, is observed to damp to one tenth of its original amplitude in 10 sec. If this damping is primarily due to sound production, what would be the frequency of oscillation of the fork in a vacuum?
• An overdamped oscillator has decaying exponentials in its description of its displacement (from the origin), one of which decays slower than the other one, so that it is (usually) the dominant term. Find specific initial conditions which cause that one to not appear in the solution, so that the other term is the important one, i.e., the only one.
• For a damped driven oscillator, calculate
1. the average power being provided to the system by the driving force,
2. the average power being loss because of the damping, and
3. the driving frequency at which this power is a maximum.
However, now find the resonant frequency for the (average value of the) kinetic energy. Explain why it is that these two frequencies are not the same.
• Consider a periodic driving function that has a period of T=2, and that has a triangular shape, up to a maximum value of 1 between t=-1/2 and t=+1/2, and is zero the remainder of the time during that period. Determine the response of a damped oscillator to this driving force, when the oscillator has a natural period of 1 and a damping factor of 1/10.
• Problem 5.22 from your text. However, in part (b), let's add the question as to what is the maximum value of the speed of the cart?
• Problem 5.28. (And if there is more time, consider problems 5.9 and 5.11.)
• A person is walking due east with constant velocity v, while standing on a turntable rotating in the counter clockwise direction with angular velocity ω. Find equations to determine his location as a function of time, as seen by someone on the ground.
• Two masses, m1 and m2 are free to move in one direction only on a horizontal, frictionless surface. The first one is attached to a rigid wall by a spring with constant k1, while the second one and first one are connected together by another spring with constant k2. Taking the displacement from equilibrium of each of them as x1 and x2, respectively, write down the equations of motion for both of them. Then take the limit as the first mass goes to zero, and use this as a method to determine the net effect of two springs hooked together, acting on just one mass.
9. 21 October:
• Consider a critically damped oscillator driven by the function f(t) = f0cos(ωt) + (1/3)f0 cos(3ωt), and determine the total motion, including transients, when the initial conditions are that x(0)=1 and v(0)=0.
• Consider a driven, damped oscillator, driven by the function which is zero for the first half of its period, and then sin(omega*t) for the second half. One can solve it either by doing a Fourier series for it, or treating it as two separate problems that must be joined together appropriately. Let's do both.
• Let a given 1-dimensional force be given by F = -kx + a/x3, and we only concern ourselves with positive values of x. Find the associated potential, and its equilibrium points. Near those equilibrium points, if a particle has an energy only very slightly larger than the minimum there, its behavior can be well approximated by simple harmonic motion. For this potential, at the equilibrium points, find the frequencies of oscillation about these minima, in terms of k and a.
• Problem 6.5 from your text.
• Find the equation of the path joining the origin to the point (1,1), in the x,y-plane, that makes stationary the integral from the origin to this point of the quantity (y')2 + yy' + y^2 , i.e., Probl. 6.9.
• Problem 6.11 from your text.
• Problem 6.23 from your text.
10. 28 October: Third Exam, on Chs. 5 and 6.
11. 4 November:
• Problem 7.41 from your text.
• Problems 7.35 and 7.50 from your text.
• Problem 7.38 from your text.
• Problems 7.37 and 7.39 from your text. In both of these, also determine the appropriate generalized momenta for the problem, and determine the Hamiltonian.
• Problem 8.1 from your text, on the conversion to center-of-mass and relative coordinates.
12. 11 November:
• Recall how to describe circles, ellipses, parabolas, and hyperbolas in Cartesian coordinates, both when their axes are aligned to the basis vectors and not, and also when the origin of the figure is at the origin or translated away from it.
• Problem 8.7
• Problem 8.11. Do this in Cartesian coordinates.
• Problems 8.16 and 8.19.
13. 18 November:
• Suppose an asteroid of mass m is caught and then released with half its earlier angular momentum, at some distance R from the center of the sun. Describe its new orbit.
• A comet is incoming on a hyperbolic orbit around our sun. The orbit has eccentricity 2 and parameter minimum distance from the sun of 0.5 astronomical units. It (first) passes the orbit of the earth on New Year's Day of 2009. How long must we wait to see it on its return trip outward?
• Determine the x,y-form of the equation for an orbit with eccentricity 1, i.e., a parabolic orbit. For such an orbit, define a parameter q such that y=cq and x = c(1-q2)/2. Determine the form of r for this situation and use that to integrate the equation for time in terms of q.
• What is the distance from the focus, i.e., from the source of our central force, when y = b? What is its (vector) velocity at that point? How long did it take to get there, from the minimum distance point?
• A spacecraft is put into an elliptical transfer orbit, to transfer from the orbit of Venus to that of Mars. What is its (vector) velocity as it passes the orbit of the earth? Assume the three planets to have circular orbits, with radii 3/4 A.U., 1 A.U., and 3/2 A.U.
• Problem 8.28
• Problem 8.29