Homework 1 Solutions - Phys 200A(Theoretical Mechanics...

Phys 200A (Theoretical Mechanics), Problem Set I
Fetter & Walecka, problem #3.2.
Done by Munirov V. R.
1) Lagrangian
The Lagrangian of the system is straightforward:
L = T - V,
L =
m
2
˙
l
2
+
m
2
⌦
2
l
2
sin
2
✓
0
- mgl cos ✓
0
.
2) Equilibrium orbit
Euler-Lagrange equations of motion:
d
dt
⇣
@ L
@
˙
l
⌘
-
@ L
@ l
= 0 ,
m
¨
l = m ⌦ 2 sin
2
✓
0
l - mg cos ✓
0
.
In equilibrium
¨
l = 0 , so we get the condition for an equilibrium circular orbit:
l
0
=
g cos ✓
0
( ⌦ sin ✓
0
)
2
, QED.
3) Stability
To consider the stability of this orbit against small displacements along the wire
we write l = l
0
+ 4 l ( Δ l ! 0 ) and put it into the equation of motion. By doing
that we get:
Δ
¨
l = ⌦ 2 sin
2
✓
0
4 l ,
which tells us that it is unstable equilibrium because coe ffi cient before 4 l is
positive.
4) Balance of force
In non-inertial rotational reference frame there are three forces acting on the
point mass: gravitational force m g acting downward, centrifugal force m ⌦
2
l sin ✓
0
acting outward from rotational orbit and reaction force of the wire m ⌦
2
sin