Download The Equilibrium of Forces: The First Law

The Equilibrium of Forces: The First Law©98
Experiment 4
Objective: To study the methods of vector addition as applied to the first law of the
equilibrium of forces.
DISCUSSION:
The first law of the equilibrium of forces states that the vector sum of the forces
acting on an unaccelerated object is zero. One way to measure the validity of this law is
to add vectorially all the forces
acting on a stationary object
F2
except one, and then to note by
what amount the resultant force
differs from the expected force.
An example is shown in Fig.1.
Here, there are three forces acting
on the object. Two of these
t F
n
a
t
l

resu
forces, F1 and F2, are added by
geometric construction to produce
the resultant force, F. This force
brium
quili
e
should be equal and opposite to F3
F3
and can be compared with it by
F1
finding the percent deviation in
their magnitudes and the percent
Figure 1. The vector addition of forces.
error between their angular
separation and 180º.
The mathematical formulation of these ideas is as follows: we let F1 and F2 be the
measured values of the magnitudes of two of the forces. Then constructing a diagram
like that shown in Fig. 1 (using the parallelogram rule for adding vectors), we measure
the magnitude, F, of the resultant. We compare this value with the magnitude F3 using
the equation for percent error:
FF
3
%Error 
 100%
(1)
F
F
Letting  be the angle between F and F3, we compare the direction of F relative to F3 by
comparing  with the theoretical value of 180 degrees. That is, we find the percent error
using:
%Error θ 
180  θ
 100%
(2)
180
Another way in which the law can be verified is to determine the algebraic sum of
the components of the forces in a given direction. As an example, the rectangular
components of the vector are shown in Fig. 2. Regardless of the direction chosen, the
sum should be zero. However, in general the sum is not zero because unavoidable errors
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y-axis
invariably appear. This sum is actually the
difference between the components in the
positive sense of the direction chosen and the
components in the negative sense. Hence,
dividing this sum by either the sum of the
components in the positive sense or the sum
in the negative sense is a measure of the
error in that direction. A complete measure
of the error is obtained when the error is
calculated once for the x-direction (chosen
arbitrarily) and again for the corresponding
y-direction.
F2y
F2
F

2

3

1
F3x
F
F1
1y
F
2x
F3
F
1x
3y
Figure 2. Vector components.
The mathematical formulation of these comparisons for the x-direction is as
follows: Defining 1 to be the angle between F1 and the x-axis in Fig. 2, we have
F  F cos α
1x
x-axis
1
(3)
1
Similar equations can be written for the other two forces. The percent error for the xdirection is
F 1x  F 2 x  F 3 x
 100%
%Error x 
(4)
F 1x
A similar expression can be written for the y-components of the forces, except that the
cosine appearing in Eq. (3) is replaced with the sine.
EXERCISES:
1. Determine the resultant force of two vectors a number of times.
a. On the equilibrium-of-forces apparatus,
i) Choose arbitrary directions for the three forces, but do not make any of the
angles between the forces right angles.
ii) Adjust the weights and/or angles so as to produce equilibrium (the ring should
be centered on the post).
iii) Record the measured values of the forces and their angular directions.
b. Graph one set of vectors whose directions and magnitudes represent the forces
from part (a). To do so, it will be necessary to choose a length-scale to represent
a given force, such as 1 cm is equivalent to 10 N.
c. Construct the resultant of two of the forces graphically using the parallelogram
rule for the addition of vectors.
i) Measure the magnitude of this resultant and determine the force it represents.
ii) Measure the angle  between the resultant and the third force.
d. Determine the percent error between the magnitude of the resultant and the
magnitude of the third force. [Use Eq. (1)]
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e. Determine the percent error between theta and the theoretical value of 180
degrees. [Use Eq. (2)]
2. Break down a force into its components.
a. Choose an arbitrarily oriented set of rectangular coordinate axes whose origin is at
the point at which the forces act. (Calculations are somewhat simplified if you let
the x-axis lie along the direction of one of the forces.)
b. With a protractor, measure the angle  each force makes with the x-axis.
c. Calculate the x- and y-components of each force using the appropriate
trigonometric functions.
d. Measure the x- and y-components of the three vectors by dropping perpendiculars
from the ends of the vectors to the coordinate axes.
e. Find the percent deviation between the graphical and analytical answers.
3. Using Eq. (4), determine the percent errors in your confirmation of the first law of
equilibrium for the x- and y-components of the forces.
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