AE250: MEASUREMENTS
Laboratory Exercise 3A:
Microprocessor-Based Data Acquisition on a Pendulum
Subject: Laboratory Handout
Prepared by: Thomas Szarek
Date exercises to be performed: 13-23 September 2004
Summary: In this laboratory, the motion of a pendulum will be investigated by
using a microcontroller-based data acquisition system (µDAQ) to record the output
of an accelerometer. The accelerometer is mounted so as to record the “centrifugal”
acceleration produced by the pendulum’s swinging motion. In this arrangement, the
accelerometer also measures gravity, the component of which acting on the accelerometer changes based on the angle of the pendulum. For this reason, the accelerometer
output will largely be used to determine when the pendulum has reached its apex
as well as the bottom of its swing. From these measurements, the period of the
pendulum can be determined.
Deliverables: Only a data sheet is required for this lab. There are four deliverables
for this lab.
• Derivation of the period for a distributed-mass pendulum using the small angle
approximation and the formulas for the period of a pendulum (both point-mass
and distributed-mass) without making the small angle approximation. You do
not have to derive the full equation of motion, just be sure to site the source
from which you found the equations.
• Complete the attached table for as many runs as were conducted.
• Plot the decay in the pendulum’s apex angle θm versus time.
• Plot the period of the pendulum T versus the pendulum’s apex angle θm and
compare this to the prediction using the four period calculations. Note that for
the small-angle approximation, T will not depend on θm .
Introduction: Galileo first began experimenting with pendulums in 1602. As legend
has it, he became intrigued by the motion of pendulums by observing the chandeliers
swinging in the cathedral of Pisa and timing their period with his pulse. Whether or
not this is true, Galileo was the first person to come to the conclusion that the period
of a pendulum was determined only by the pendulum’s length [1]. Later Christian
1
Huygens went on to develop the equation governing a simple pendulum and invented
the pendulum clock [2].
The development of the equation for a point-mass pendulum undergoing small
swings is presented below.
Figure 1: Forces Acting on Pendulum.
Figure 1 shows a pendulum at some angle θ. In this position, the pendulum is
subjected to a moment that will cause the pendulum to fall towards the center. The
moment is defined to be the cross product of the vector from the point of rotation to
the center of the mass and the force vector. In this case, the force vector will always
point down as gravity always acts downwards. The resultant becomes:
τ = ~` × m~g
τ = −`mg sin θ
(1)
(2)
where ` is the distance from the point of rotation to the center of mass. It should be
noted that for a point mass, this distance is simply the “length” of the pendulum.
2
For a real pendulum with distributed mass, this distance needs to be calculated, and
the equations will become more complicated, although not unbearable.
From Newton’s Second Law, this unbalanced torque will lead to an acceleration
that will be retarded only by the inertia of the system. In the case of rotation, the
inertial takes the form of the mass moment of inertia. If we again assume a point
mass, the equation becomes:
τ = I θ̈
τ = m`2 θ̈
(3)
(4)
where θ̈ is simply the second derivative of θ with respect to time, or the angular
acceleration. It should be noted that for a distributed mass system, the inertia will
take a different form, but will not prevent a solution of this form.
Combining these two equations for the torque results in:
m`2 θ̈ = −`mg sin θ
g
θ̈ + sin θ = 0
`
(5)
(6)
This gives us a differential equation that is unfortunately non-linear. The sine term
makes this equation impossible to solve easily; however, realizing that for small angles
that sin θ ≈ θ, we can rewrite the equation in a solvable form:
g
θ̈ + θ = 0
`
(7)
(8)
with four solutions:
r
g
t
l
r
g
t
θ = −C sin
l
r
g
θ = C cos
t
l
r
g
t
θ = −C cos
l
θ = C sin
(9)
(10)
(11)
(12)
The exact form of the solution and the value of C depends on the initial conditions, for
example, where in the pendulum’s cycle is t defined to be zero, what is the maximum
3
release angle, and what direction is defined as a positive θ. By differentiating any of
these equations twice and plugging back into the original equation, the validity of the
form of the solution can be tested.
What is important to note is that the pendulum will go through one cycle when
the term inside the sine or cosine goes from zero to 2π. This means that the period
of the motion can be found to be:
r
g
T = 2π
l
s
T = 2π
(13)
l
g
(14)
The confirms Galileo’s observation that the period of a simple pendulum depends only
on the length of the pendulum (and gravity, but unless you go to another planet, that
is constant).
For this lab, you should derive the period using this method, but do not assume
a point mass. Derive the equation such that given a slender rod of certain length
and mass and a circular disk of certain radius and mass, you can calculate the actual
center of mass and mass moment of inertia.
You should also search either in a book or on the internet and find the solution
to the full equation of motion without making the small angle approximation. It will
take the form of the simplified versions, but multiplied by a modifying factor that
accounts for the peak angle of the oscillation.
Accelerometer: The accelerometer is a spring-mass system that responds to accelerations in one direction. The mass deflects an amount proportional to the acceleration. This deflection is detected electronically and conditioned so a voltage is output
proportional to the acceleration.
Like any spring-mass system, the accelerometer is subject to gravity, meaning
that if the accelerometer is held with the sensitive axis pointing down, it will measure
an acceleration of 9.81 m/s2 . If the sensitive axis is held horizontal, it will measure
zero acceleration, and if the sensitive axis is held pointing up, it will measure an
acceleration of -9.81 m/s2 .
This is useful because it provides and easy method with which to calibrate the
device since it can be subjected to well known accelerations; however, in the operation
on the pendulum, it will measure not only the “centrifugal” acceleration, but also
some component of gravity that depends on the angle of pendulum.
a = r θ̇2 + g cos θ
4
(15)
Unless the relationship between θ and θ̇ is known, which it is difficult to know without
simplifying the problem, it is impossible to determine from a given measurement, what
component is from “centrifugal” acceleration and what component is from gravity.
It is possible; however, to glean some information from the output, specifically
when the pendulum is at its apex and when it is at the bottom of its swing. At the
apex, the angular velocity is zero. This means that the centrifugal acceleration is also
zero. There are no negative centrifugal accelerations. Also at the apex, θ will be at
its maximum, so the cos θ will be at a minimum. By looking for the minimums in
the time series, the apex points can be defined and the angle θm can be calculated.
At the bottom of the swing, the angular velocity will be a maximum, and therefore
the centrifugal acceleration will also be a maximum. By definition, θ will be zero,
so cos θ will be one, and thus the gravity component will be maximum at this point.
By finding the maximum locations in the time series, the bottom of the swing can be
defined.
With this information, it is possible to obtain the period of the pendulum as a
function of time with the period determined as the average over a quarter cycle, and
the maximum swing angle as a function of time as measured at each extreme of the
cycle.
References:
1 “From Galileo’s pendulum to the Global Positioning System.” Online. Indiana
University Media Relations. http://newsinfo.iu.edu/news/page/normal/1427.html.
7 September 2004.
2 Omar, Usuff. “Pendulum Photography.” Online. http://bpresent.com/usuff/huygensfoucault.php. 7 September 2004.
5
Names:
Initial Release Angle:
NOTE: It is not possible to define the release point. Begin with the bottom of the
first swing at the 1/4 cycle point. The acceleration is not needed at the bottom of
the cycle (1/4 and 3/4 points). The period calculated will be the average over the
previous quarter cycle. The Maximum Angle should be calculated at the 1/2 and full
cycle points and the average of those taken at the 1/4 and 3/4 points.
Cycle
0
1/4
1/2
3/4
1
1 1/4
1 1/2
1 3/4
2
2 1/4
2 1/2
2 3/4
3
3 1/4
3 1/2
3 3/4
4
4 1/4
4 1/2
4 3/4
5
5 1/4
5 1/2
5 3/4
Time
(s)
-
Acceleration
(m/s2 )
-
6
Period
(s)
-
Maximum Angle
(deg)
-
Cycle
6
6 1/4
6 1/2
6 3/4
7
7 1/4
7 1/2
7 3/4
8
8 1/4
8 1/2
8 3/4
9
9 1/4
9 1/2
9 3/4
10
10 1/4
10 1/2
10 3/4
11
11 1/4
11 1/2
11 3/4
12
12 1/4
12 1/2
12 3/4
13
13 1/4
13 1/2
13 3/4
14
Time
(s)
Acceleration
(m/s2 )
Period
(s)
Maximum Angle
(deg)
-
-
-
-
7
Names:
Initial Release Angle:
NOTE: It is not possible to define the release point. Begin with the bottom of the
first swing at the 1/4 cycle point. The acceleration is not needed at the bottom of
the cycle (1/4 and 3/4 points). The period calculated will be the average over the
previous quarter cycle. The Maximum Angle should be calculated at the 1/2 and full
cycle points and the average of those taken at the 1/4 and 3/4 points.
Cycle
0
1/4
1/2
3/4
1
1 1/4
1 1/2
1 3/4
2
2 1/4
2 1/2
2 3/4
3
3 1/4
3 1/2
3 3/4
4
4 1/4
4 1/2
4 3/4
5
5 1/4
5 1/2
5 3/4
Time
(s)
-
Acceleration
(m/s2 )
-
8
Period
(s)
-
Maximum Angle
(deg)
-
Cycle
6
6 1/4
6 1/2
6 3/4
7
7 1/4
7 1/2
7 3/4
8
8 1/4
8 1/2
8 3/4
9
9 1/4
9 1/2
9 3/4
10
10 1/4
10 1/2
10 3/4
11
11 1/4
11 1/2
11 3/4
12
12 1/4
12 1/2
12 3/4
13
13 1/4
13 1/2
13 3/4
14
Time
(s)
Acceleration
(m/s2 )
Period
(s)
Maximum Angle
(deg)
-
-
-
-
9
Names:
Initial Release Angle:
NOTE: It is not possible to define the release point. Begin with the bottom of the
first swing at the 1/4 cycle point. The acceleration is not needed at the bottom of
the cycle (1/4 and 3/4 points). The period calculated will be the average over the
previous quarter cycle. The Maximum Angle should be calculated at the 1/2 and full
cycle points and the average of those taken at the 1/4 and 3/4 points.
Cycle
0
1/4
1/2
3/4
1
1 1/4
1 1/2
1 3/4
2
2 1/4
2 1/2
2 3/4
3
3 1/4
3 1/2
3 3/4
4
4 1/4
4 1/2
4 3/4
5
5 1/4
5 1/2
5 3/4
Time
(s)
-
Acceleration
(m/s2 )
-
10
Period
(s)
-
Maximum Angle
(deg)
-
Cycle
6
6 1/4
6 1/2
6 3/4
7
7 1/4
7 1/2
7 3/4
8
8 1/4
8 1/2
8 3/4
9
9 1/4
9 1/2
9 3/4
10
10 1/4
10 1/2
10 3/4
11
11 1/4
11 1/2
11 3/4
12
12 1/4
12 1/2
12 3/4
13
13 1/4
13 1/2
13 3/4
14
Time
(s)
Acceleration
(m/s2 )
Period
(s)
Maximum Angle
(deg)
-
-
-
-
11