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 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 forces, F1 and F2, are added by geometric construction to produce the resultant force, F. This force should be equal and opposite to F3 and can be compared with it by finding the percent deviation in their magnitudes and the percent 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:

(1)

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:

(2)

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 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.

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
(3)

Similar equations can be written for the other two forces. The percent error for the x-direction is

(4)

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