## S-1 VECTOR ADDITION OF FORCES
(1) Study static equilibrium under the action of several concurrentforces. (2) Test whether the forces on a body at rest add to zero. (3) Study how errors in vectors propagate.
Force table, three weight hangers, metric weight set, 17 x 11 or 17 x 22" drawing paper.
Read the chapters on vectors and components of vectors in any physics textbook.
You can avoid unnecessary multiplications by 9.8 m/s
In this experiment you will study the special case of a body at rest under the action of only three forces.
(1) The vector polygon of all forces acting on the body in static equilibrium is closed. The sum of the vectors representing the forces is zero. (2) The sum of the components of force along any axis is zero.
Read this section before beginning section 7, Procedure.
Take care to adjust the pulleys so the strings pass straight over them, ensuring minimum friction. Use enough weight on the hangers so the strings are nearly parallel to the table and do not sag much. Why is this important?
When you achieve equilibrium of the body, the directions of the forces lie along the direction of the strings. In each situation, adjust the weights and angles so that the ring is centered on the table, and is not touching the center post. Record the amount of weight on each hanger, and read the direction of its string on the table's scale.
Each different situation may be recorded on a separate sheet of drawing paper. The analysis for each situation is then done directly on that work sheet. Use 200 to 500 gram weights on the hangers. Avoid situations where all hangers have
the same weight. Try for a representative, general case.
Fig. 3. illustrates the addition of three vectors by the polygon method. The vectors
may be assembled into the polygon in any order, since vector addition is commutative.
In the case of
static equilibrium, which you are investigating, the vectors will very likely add to nearly zero,
so that any resultant
On a clear area of your work sheet, transfer the forces onto a vector polygon and test
whether the polygon closes. It probably won't close perfectly. Why not? What is the size and
direction of the additional "error" vector required to close the polygon? Is its size consistent
with the error caused by error in each of the individual forces? This last question will require
considerable analysis, and a consideration of how to do an error propagation calculation for
B. Its head is at coordinate (16.2,4.7) and its tail is at (6.5,9.8). Its x
component is therefore 16.2 - 6.5 = 9.7, and its y component is 4.7 - 9.8 = -5.1. Notice
how the y component came out negative. The vector is pointing "downward" (negative) with
respect to the y axis, but "forward" (positive) with respect to the x axis. A. (b) Find the components of A+B. (c) Find the components
of vector C. (d) Find the components of A+B+C.Good "bookkeeping" is essential when doing these problems. For example, the following table keeps track of the numbers of exercise 1. We have only filled in part of it: you may fill in the rest.
The size (magnitude) of a vector is always positive, and may be found by using the
Pythagorean theorem if the components are known. The angle of a vector is conveniently
specified with respect to the positive x axis, measured counterclockwise. If the
components are known, the angle is arctan(V Note that your electronic calculator always returns an angle less than 90°, which you must convert to correctly indicate the
quadrant in which the angle lies. Obviously the calculator cannot do more for you
because of the ambiguity of sign: V
(1) If the weights of the weight hangers were exactly the same could their weights have been neglected in the computations? Explain. (2) One form of apparatus for this experiment has a special circular table with degree markings around its rim. Three forces are applied to strings attached to a small round metal ring around a center post. The center post keeps the ring centered while the system is being balanced. When balance is achieved the ring is centered on the post, but not touching the post. The post can then be removed. Was the ring in equilibrium during the time when the weights were being balanced, while the ring was still touching the post? (3) Can one place three forces of 50, 75, and 150 grams at such angles that they will produce equilibrium? Explain your answer, and try to make a general statement about this situation. (4) Where on earth can one travel 10 miles north, then 10 miles west, then 10 miles south and end up at the starting point? Why doesn't this seem to obey the laws of vector addition. (5) One answer to question (6) was easy to find, but there are other answers. In fact, there are an unlimited number of other starting points on the surface of the earth from which this may be done, and they all have a special relation to each other. Make a general statement describing the location of these points. (6) [Cioffari] The x and y cartesian components of a certain force are measured and found to be 68 ± 3 and 42 ± 2 N respectively. Calculate the direction and magnitude of this force, expressing your result to the proper number of significant figures and showing the error in both magnitude and angle. Note that the error in the latter is obtained from the error in its tangent. Devise a procedure for doing this and show your work. Text and drawings ©: 1997, 2004 by Donald E. Simanek. |