PHY4222: CLASSICAL MECHANICS
TEST II
April 6, 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.
1. A charged bob (mass m, charge q > 0) is suspened in a uniform and time-independent electric field E, as shown
in Fig. 1a.
a) Write down the Lagrangian in terms of angle φ (not assuming that it is small!) and derive the equations
of motion. [10 points]
T =
1 2 2
m` φ̇
2
Electric field produces a force on the bob
FE = qE
Corresponding potential energy
UE = −
Z
dxFE = −qEx = −qE` sin φ
Gravitational potential energy
Ug = −mg` cos φ
Alltogether,
L = T − U E − Ug =
1 2 2
m` φ̇ + mg` cos φ + qE` sin φ
2
∂L
d ∂L
= m`2 φ̈ =
= −mg` sin φ + qE` cos φ
dt ∂ φ̇
∂φ
b) Determine the equilibrium position of the bob (again, not assuming that φ is small!)[10 points]
φ̈ = 0 → mg` sin φ0 = qE` cos φ0
tan φ0 = qE/mg
φ0 = arctan (qE/mg) .
c) Find the frequency of small oscillations about the equilibrium position.[10 points]
Using (1), we can re-write the RHS of the Euler’s equation as
∂L
cos φ0
= −mg` sin φ + qE` cos φ = −qE`
sin φ + qE` cos φ
∂φ
sin φ0
cos φ0
qE`
= −qE`
sin φ − cos φ = −
[sin φ cos φ0 − cos φ sin φ0 ]
sin φ0
sin φ0
qE`
= −
sin (φ − φ0 )
sin φ0
(1)
2
Expanding the right-hand side of the equation of motion about in θ = φ − φ0 (|θ| 1),we obtain
m`2 θ̈ = −
qE`
θ.
sin φ0
Therefore,
ω02 =
qE
m` sin φ0
A little bit of trigonometry and algebra leads to
1 + tan2 φ0 = 1/ cos2 φ0
p
sin φ0 =
g
`
ω02 =
1 − cos2 φ0 = p
q
1 + tan2 φ0
2
1 + (qE/mg)
g 2
ω0 =
tan φ0
`
+
qE
m`
2 !1/4
.
d) Expand the equations of motion in a small deviation from the equilibrium position, retaining the leading
non-linear term. [10 points]
Expanding ∂L/∂φ further, we obtain
θ3
qE`
qE`
∂L
θ−
=−
sin θ ≈ −
∂φ
sin φ0
sin φ0
6
e) Suppose that, in addition to the time-independent electric field, the pendulum is also driven by a periodic
force F = F0 cos ωt. What frequencies of the external force would lead to a resonance in this system? [10
points]
The θ3 term generates a fractional resonance at ω = ω0 /3.
2.
a) Calculate the tensor of inertia of a homogeneous cube (mass M , side a) in a coordinate system with one
corner as the origin and the edges as the coordinate axes. [25 points]
Iij = ρ
Z
I11 = I22
dV δij r2 − ri rj
Z a
Z
= I33 = ρ
dx1
0
a
0
I12 = I21 = I13 = I31 = I23 = I32
a
3
a3
2
a
+
= M a2
dx3 x22 + x23 = ρa
3
3
3
0
2 2
Z a
Z a
Z a
a
1
dx3 x1 x2 = −ρa
dx2
dx1
= −ρ
= − M a2
2
4
0
0
0
dx2
Z

2/3 −1/4 −1/4
Iˆ = M a2  −1/4 2/3 −1/4 
−1/4 −1/4 2/3

b) Suppose that this cube is initially in a position of unstable equilibrium with one edge in contact with a
horizontal plane (see Fig. 1a). If tips over and falls (without sliding), find the angular frequency of the
cube when one face strikes the plane. [25 points]
Initial energy
√
E = Ui = M ga 2/2
3
E
φ
Figure 1a
Figure 1b
FIG. 1:
Final kinetic energy
Tf =
I11 ω 2
12
1
=
M a2 ω 2 = M a2 ω 2
2
23
3
Final potential energy
Uf = M ga//2
Energy conservation
Ui = T f + U f
Tf = U i − U f
√
1
2 − 1 /2
M a2 ω 2 = M ga
3
g
3 √
.
ω2 =
2−1
2
a
r
r
r g
3 √
g
≈ 0.788
ω =
2−1
2
a
a