Kater's Reversible Pendulum Advanced Lab - Earlham Computer ...

Kater’s Reversible Pendulum
Advanced Lab
Seth Hopper
December 8, 2003
Abstract
I measured the local acceleration due to gravity using Kater’s Reversible Pendulum. The
pendulum could be suspended in two positions, and when the periods of the two positions were
equal, the setup allowed for a simple measurement of g. I calculated g within 0.1% of the
accepted value for Richmond, IN.
Contents
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1 Introduction
The local acceleration due to gravity is perhaps the most commonly used constant in all physics
calculations. It can be measured many different ways, some with more accuracy than others. The
method I used in this lab involved a reversible pendulum, developed by Kater in 1817.
Kater’s reversible pendulum is a physical pendulum with movable masses on either end, as
depicted in Figure
2 Theory
In order to motivate the use of the reversible pendulum, we consider the scenario depicted in Figure
2. We assume that the masses M 1 and M 2 have been positioned as to make the periods of the two
Figure 1:
orientations of the pendulum equal. Here the physical pendulum is suspended from a knife edge
at point P 1 . The center of mass is at point C, and the distance between these two points is, as
displayed, h 1 . Let the total mass of the pendulum be M and the moment of inertia about point P 1
2

e I P1 . Then, the torque on the pendulum from gravity is
τ = I P1
d 2 θ
dt 2 = −Mgh 1 sin(θ). (1)
For our purposes, let us only consider cases when θ is small; then we can make the approximation
sin(θ) = θ. Then the period for this physical pendulum becomes
√
I P1
T = 2π . (2)
Mgh 1
Now, we recognize that from the parallel axis theorem that
I P1 = I C + Mh 2 1 (3)
where I C , is the moment of inertia about the center of mass of the pendulum. At this point we
define the radius of gyration k of the pendulum about C to be
k = √ I C /M. (4)
Hence, Equation
. We assume that the masses M 1 and M 2 have been positioned as to make the periods of the
two orientations of the pendulum equal. Here the physical pendulum is suspended from a knife
edge at point P 1 . The center of mass is at point C, and the distance between these two points is,
as displayed, h 1 . Let the total mass of the pendulum be M and the moment of inertia about point
P 1 be I P1 . Then, the torque on the pendulum from gravity is
τ = I P1
d 2 θ
dt 2 = −Mgh 1 sin(θ). (5)
For our purposes, let us only consider cases when θ is small; then we can make the approximation
3

Figure 2:
sin(θ) = θ. Then the period for this physical pendulum becomes
√
I P1
T = 2π . (6)
Mgh 1
Now, we recognize that from the parallel axis theorem that
I P1 = I C + Mh 2 1 (7)
where I C , is the moment of inertia about the center of mass of the pendulum. At this point we
define the radius of gyration k of the pendulum about C to be
k = √ I C /M. (8)
Hence, Equation
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3 Procedure
The key to measuring the acceleration due to gravity with Kater’s reversible pendulum is finding
a positioning where the period is identical when the pendulum is flipped vertically and suspended
from the other knife edge. In finding this position, I used the following equipment.
3.1 Equipment
1. EM-12 Kater’s Reversible Pendulum with stable wall mount
2. PASCO Scientific Photogate
3. Apple Computer running Science Workshop
4. Lab Jack Adjustable Stand
5. 2 Meter Stick
6. Calipers
7. Straight Blade Screwdriver
3.2 In General
I set up the equipment as shown in Figure
Figure 3:
The equipment setup for measuring periods of the Kater pendulum.
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There are many different orientations that will yield equal upright and inverted periods for the
Kater pendulum. In fact, for every position of M 1 there is one position of M 2 where the periods will
be identical. I placed M 1 at one position and moved M 2 , making measurements every 5 cm until
the periods were equal. Then I took more accurate measurements of the position of M 2 , moving it
only 2 cm at a time in the range where I found the periods to be equal.
3.3 The Specifics
I hung the pendulum from the wall mount with M 1 at the top. Using the adjustable stand, I placed
the photogate directly underneath the bottom of the pendulum so that when the pendulum was at
its minimum it blocked the signal between the two sensors of the photogate. When the pendulum
would oscillate the photogate would report every time the signal was interrupted, which was every
half period. These signals were sent to the computer running Science Workshop software, which
measured the time intervals between interrupts. These intervals were displayed on the screen, and
I observed them for many different positions of M 1 2 .
I started M 2 78 cm from the bottom of the pendulum (as oriented in Figure
4 Results and Conclusions
The following graphs display the periods of the Kater reversible pendulum in both orientations as
a function of the position of M 2 . The intersection of the lines indicates the position where the
periods for the different orientations of the pendulum are equal. The second graph is basically a
zoom of the first graph, and I see that the intersection occurs at 2.006 s.
The formula for g comes
from Equation
How accurate is this Well, a formula for the local acceleration due to gravity is as follows.
g = 9.780 318 4(1 + 0.005 302 4 sin 2 L − 0.000 005 9 sin 2 2L) − 3.086 × 10 −6 H (9)
Here L is the latitude and H is the height above sea level. For Richmond, IN the latitude is 39.83 ◦
1 The computer actually displayed whole periods, not half periods, so it only output data every other signal
interrupt.
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2.06
2.04
2.02
”Up”
”Down”
2
1.98
Period (s)
1.96
1.94
1.92
1.9
1.88
1.86
0
0.1
0.2
0.3 0.4 0.5
M2 Position (cm)
0.6
0.7
0.8
2.06
2.05
”Up Zoom”
”Down Zoom”
2.04
Period (s)
2.03
2.02
2.01
2
0
0.02
0.04
0.06 0.08 0.1
M2 Position (cm)
0.12
0.14
0.16
and the elevation is 286.2 m. Hence, the local acceleration due to gravity should be
g = 9.780 318 4(1 + 0.005 302 4 sin 2 (39.83) − (10)
5.9 × 10 −6 sin 2 (2 · 39.83)) − 3.086 × 10 −6 · 286.2
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g = 9.801 m/s 2 . (11)
The acceptable range for my result is
9.637 m/s 2 ≤ g ≤ 9.985 m/s 2 . (12)
Clearly the accepted value for g falls well within this range. In fact, my result is barely .1% off
from the accepted value. Perhaps my range of uncertainty is too large. However, I do believe that
the uncertainty in H is accurate. It would not surprise me, on the other hand, if the uncertainty
in T were smaller than I estimated.
In conclusion, I conclude that the data I took from the reversible Kater pendulum gave conclusive
results that concur with those taken by other people in this world. The possible reasons for
differences between my results and the accepted values include that my elevation and latitude are
probably not exactly what I found by searching on the Internet. Also, there could be local mineral
deposits that could alter the local value of g from what would be accepted. I ignored any friction,
and also made the approximation that sin(θ) = θ, but I doubt if either of these played a role in
any inaccuracies. If I had to guess, and I do, I think the uncertainties in T and H were mostly
responsible for the majority of my discrepancy.
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