October 2, 2003
Physics 16
Problem Set 2
1.
Form the derivation in our textbook, we know that the general solution for a damped
and driven harmonic oscillator with damping force bv and driving force F
0
cos(
wt
) is:
In this problem, b=mc, c=6w/5, and w
d
= w.
We can easily solve for some of the constants in this equation:
3
2
5
b
w
m
γ
=
=
2
2 2
2
(
)
(2
)
2
R
w
w
w
w
=

+
=
2
2
1
1
cos (
)
cos (0)
2
w
w
R
π
φ



=
=
=
2
2
2
2
16
25
w
w
Ω =

= 
Note that c
2
< 0, so the system is underdamped.
Thus, using the solution in the textbook,
we can replace the above equation with:
2
0
16
( )
cos(
)
cos(
)
25
t
F
w
x t
wt
e
C
R
ϕ

=

+
+
All that remains is to solve for C and c.
Our initial conditions are x[0]=0 and x'[0]=0.
Solving simultaneously with Mathematica, one solution is:
0
2
25
24
F
C
w

= 
0
( )
cos(
)
(
)
t
t
t
F
x t
wt
e
Ae
Be
R

Ω
Ω
=

+
+
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For each of the following, I determined whether the force could be derived from a
potential by finding its gradient and seeing whether or not it was equal to the zero vector.
I found the potentials by guessandcheck, not by path integrals.
a.
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 Force, Kinetic Energy, Momentum, Cos, initial conditions

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