Take the log of both sides and then expand according to rules for logs.
y = y
o
e
kt
log y = log (y
o
e
kt
)
= log y
o
+ log (e
kt
)
9

At this point we observe a significant difference.
If we are using natural logs (base e) rather
than common logs (base 10) then we can make a substantial additional simplification.
Recall that the symbol "ln" is used to indicate natural logs in circumstances when it is
important to know the base of the log calculation.
Then:
ℓ
n y =
ℓ
n y
o
+
ℓ
n (e
kt
)
=
ℓ
n y
o
+ kt
So!
If the relationship between y and t is exponential, then a plot of
ℓ
n y vs. t (NOT vs.
ℓ
n t)
will give a straight line whose slope provides the value of k and whose y- intercept (at t = 0)
provides y
o
.
(A)
Now add a 3
rd
column (neatly) to the data table in section 3(A).
The new column should contain
the values of
ℓ
n V for each of the 10 values of V.
I.
Graph your data for
ℓ
n V vs. t on the sheet of Cartesian paper following
this page.
ℓ
n V is dependent, t is independent.
Sketch the best straight line.
II.
Find Vo and the slope k (k = -
= -1/ T = -1/ T
e
) from your straight line.
Show your calculation for K below.
Circle y
o
on your graph.
V
o
=
________
_____
K =
________
_____
=
________
_____
T
e
= T =
________
_____
value
units
III.
Compare your values of V
o
and
from part II above to the values of V
o
and
from parts 3(C) and (G).
3(C) and (G):
V
o
= ________
=
________
part 2 above:
V
o
= ________
=
________
Comment:
10

11

12

PH262
Simple Harmonic Motion and the Pendulum
Lab
# ____
Name ______________________________________________ Date _________
Lab Partner(s) Name ___________________________________________________________
III.
EQUIPMENT LIST
Tall Lab Stand
C-Clamp
15” Spring
Pendulum Clamp
Cardboard Dampener
Assorted Hooked Masses
Spring Scale (in Newtons)
Cotton String (≈ 1.0m long)
IV.
BACKGROUND INFORMATION
III. EXPERIMENTAL PROCEDURE
1.
Simple Harmonic Motion - Mass/Spring System
Find a support base and rod.
Use a "C" clamp to fasten the
stand to a lab table.
Attach a pendulum clamp near the top of
the rod.
Attach the spring to the pendulum clamp to the
innermost support point. Hang a 1 kg mass from the spring.
The point where the weight is hanging at rest is the static
equilibrium point, y = 0.
Pull the weight down about 10 -15 cm
from equilibrium and release it.
Measure Δy before you release
the weight.
Δy = y
0
= ________
_____
value
units
(A)
Measure the period,
T
- let the spring/mass oscillate about 10
complete cycles, find the total t using a stop watch and
calculate
T
.
Repeat this measurement at least once for
consistency.
Use
T
to find
f
(frequency) and
ω
(angular frequency).
T
= ________
___
f
=
________
___
Note:
2 or 3
ω
= ________
___
significant figures
(B)
Measure K (spring constant) for your spring.
Don't spend a lot of time on this; it
is easy to accurately determine K with no more than 3 values of F vs. Δy or F vs.
Δx.
(You can use F
g
= mg or a spring scale.)
K = ________
___
m = ________
___
13

(C)
Calculate ω
C
= (k/m)
1/2
and compare to your experimental value in part (A).
Use
the space below to compare and comment on the values of ω
A
and ω
C
.

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- Fall '09
- WILSON
- Physics, Alternating Current, Resistor, ........., Lab Partner