PHYS 218  SOLUTION TO ASSIGNMENT 1
02 Feb 2007
By Chung Koo Kim
email : [email protected]
1. Eﬀect of gravity on a hanging spring
(a) One arrow pointing down, and another one up, representing weight (=
mg
) and restoring force of spring (
=
k
(
l

l
0
)), respectively. (Diagram omitted)
(b) From Newton’s 2nd law for vertical direction,
mg

k
(
l

l
0
) = 0, or
l
=
l
0
+
mg
k
(c) If the mass is pulled down by y from the new equilibrium, we get, from
the result of (b),
F
=
m
¨
y
=

k
(
l
+
y

l
0
) +
mg
=

ky
thus the equation of motion has the same form.
Then what is diﬀerent in the presence of gravity? Nothing except for the
center of oscillation. Center position, is incorporated as the constant term
of general solution (C in
y
(
t
) =
A
sin(
ωt
)+C), and it is determined by the
equilibrium of force. Neither amplitude nor frequency change.
2. Pendulum
(a) Eq. of motion can be found using either force or torque. Using the latter,

mglsinθ
=
ml
2
¨
θ
¨
θ
=

g
l
sinθ
∼ 
g
l
θ
ω
0
=
r
g
l
,
T
= 2
π
s
l
g
(b) From the result of (a), for
l
∼
20
m
,
T
∼
9
.
0
s
(c) If we assume that the number of pulse per minute is roughly 60 (or N in
general) times, the actual time interval between each pulse is 1 sec (or 60/N).
Therefore if Galileo measured the period by counting his pulse, the precision
would be order of one interval. He could have improved the precision by counting
his pulse over many periods and divide by total number of oscillation.
1
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View Full Document(d) The amplitude of a damped harmonic oscillator with a damping constant
γ
is given by
A
(
t
) =
A
0
e

γt/
2
from the general solution
θ
(
t
) =
A
0
e

γt/
2
cos(
ω
0
t
+
φ
). So if the amplitude
decreased by a factor of 2 in 30min, we solve
A
0
/
2 =
A
0
e

γt/
2
for
γ
to get
γ
=
2
ln
2
t
= 7
.
7
×
10

4
s

1
and
Q
=
ω
0
γ
=
t
2
ln
2
r
g
l
∼
909
(e) For
Q
= 10
10
, the damping constant
γ
is :
γ
=
ω
0
Q
= 7
×
10

11
s

1
Then, from the amplitude relation, we get
A
2006
=
A
1581
e

γt/
2
∼
0
.
627
A
1581
,
ﬁnding that the amplitude is still about 62.7% of the original one even after 425
years have passed. Here we see that higher Q yields less damping, or higher
stability of operation.
3. Energy of SHO
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 Spring '08
 WITTICH,P
 Force, Gravity, Normal mode, damped harmonic oscillator, Chung Koo Kim

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