Name:
UF ID:
PHY4222: CLASSICAL MECHANICS
FINAL EXAM
30 April, 2007
Instructor: D. L. Maslov
maslov@phys.ufl.edu 392-0513 Rm. 2114
Please help your instructor by doing your work neatly.
Every (algebraic) final result must be supplemented by a check of units. Without such a check, no
more than 75% of the credit will be given even for an otherwise correct solution. On the other hand,
an answer obtained just by the dimensional analysis may give you up to 25% of the credit.
• Print your name in BLOCK LETTERS and UF ID in the top left corner of this page
• Mark 2 out of 3 problems for grading using the table below. Each problem is worth 50 points.
No partial credit will be given for the problem which you did not choose to be graded.
123
• Start each problem on a new page.
• Do not turn in sheets not relevant to the problems you chose to be graded.
• Do not write below this line
—————————————————————————————————————–
For instructor’s use only
part max actual
a
15
1 b
15
c
20
part max actual
a
15
2 b
15
c
20
part max actual
a
15
2 b
15
c
20
Total:
2
1.
A bead of mass m slides without friction on a wire which connects the origin (O) with the point
P . The bead starts at the origin with an initial speed v0 .
a) Obtain an integral equation for the time of descent. [15 points]
b) Use the calculus of variations to minimize the time of descent and obtain the differential
equation for the shape of the wire. [15 points]
c) Find the shape of the wire in a parametric form x = x (θ) , y = y (θ). Sketch this shape.
[20 points]
3
2.
The point of support of a simple pendulum of mass m and length ` moves uniformly in a vertical
circle of radius R with a constant angular velocity ω.
a) Construct the Lagrangian for this system. [15 points]
b) Derive the equations of motion. [15 points]
c) Linearize the equation of motion keeping only the leading
term in φ, and show that the
q
system exhibits both an ordinary resonance at ω = g/` and a parametric resonance at
q
ω = 2 g/`. Hint: Apply the trigonometric identity cos (φ + ωt) = cos φ cos ωt − sin φ sin ωt
in the equation of motion, obtained in part b), and then linearize with respect to φ. [20
points]
4
3.
A comet of mass m in moving in a central force potential U (r) = −k/r with angular momentum
L. Neglect the difference between m and the reduced mass.
a) Find the radius of a circular orbit in terms of k, m, and L. [15 points]
b) Now consider a parabolic orbit in the same potential with the same m and L. Show that
the distance of the closest approach on this parabolic orbit is half the radius of the circular
orbit. [15 points]
c) Show that at the points where the parabolic
orbit intersects the circular one, the speed of
√
the comet on the parabolic orbit is 2 times the speed on the circular orbit.[20 points]
See next page for figures ⇒
R
O
l
P
m
Problem 2
Problem 1
Problem 3