## S-5 FORCES AND TORQUES
(1) Study static equilibrium under the action of several forces. (2) Test whether the forces on a body at rest add to zero. (3) Test whether the torques on a body at rest add to zero.
Drawing board, four pulleys with table edge clamps, four weight hangers, metric mass set (colloquially called "weights"), fiberboard "body" with four holes and attached strings, ruler, draftsman's triangle, small mirror, 17 x 22" drawing paper.
Read the chapters on vectors and torques in any physics textbook.
You can avoid unnecessary multiplications by 9.8 m/s
In two-dimensional problems the directional sense of a torque may be clockwise or counter-clockwise. The sense of the torque is easily determined by considering what direction the lever arm would rotate if it were a stick pivoted at the center or torques and "pulled" by the force. Conventionally, clockwise torques are assigned a negative sign, and counterclockwise torques are positive. In this experiment you will study the special case of a body at rest, having zero velocity.
Such a situation is called
(1) The vector polygon of all forces acting on the body 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. (3) The sum of all torques acting on the body is zero, about any chosen axis.
Read this section before beginning section 7, Procedure.
In Fig. 2A: f fiberboard rectangle
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. These directions must be transferred accurately to the drawing paper. Do this with the aid of a small mirror. Place the mirror face up on the table under a string, then sight from above, moving your head until the image of the string seen in the mirror is directly below the string and therefore hidden by the string. With your eye in this position, make a pencil mark directly under the string at each end of the mirror (so the mark is also hidden by the string). This procedure ensures that the experimenter's eye is directly above the string and that the line of sight is perpendicular to the mirror, thereby eliminating errors due to parallax. Remove the mirror and draw a line through the points with a ruler. The line will lie accurately along the direction of the string and therefore records the direction of the force due to that string.
Do this for each string, then sketch the position of the fiberboard rectangle onto the paper. Remove the drawing paper. Subsequent analysis is done directly on this paper. Draw the lines of action of the forces on the sheet. Now choose a scale to represent forces as lengths. For example, you might choose 1 cm to represent 10 gm. Choose a scale large enough to give good precision, while still keeping the diagram within the sheet of paper.
Each different situation is recorded on a separate sheet of drawing paper. The force and torque 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.
For the three force case, carry out analyses (1) and (2). For the four force case, do analyses (1), (2), and (3).
Fig. 5. illustrates the addition of three vectors by the polygon method. Note particularly
that the original vectors do not need to have their tails at the same point. 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 their 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
To gain the maximum instructional value from this exercise, we ask you do to it two ways: (1) (2)
(1) If the weights of the weight hangers were exactly the same could their weights have been neglected in the computations? Explain. (2) Devise a mathematical proof of the following proposition: "When only three forces act on a body in equilibrium, they are either all parallel, or they pass through a single point." (3) Prove mathematically that if the sum of torques on a body is zero about some
particular axis, it must be zero about Text and drawings ©: 1997, 2004 by Donald E. Simanek. |