A common contrivance using this principle is the spring balance (Fig. 55), with which all are familiar, as ice scales, meat scales, postal scales, etc. The object which changes shape in this device is a coiled spring contained in the case of the instrument. The balance is so constructed that when the spring is pulled out as far as possible it has not reached its limit of elasticity, since, if the spring were stretched so as to exceed its elastic limit, the index would not return to its first position on removing the load. (See Arts. 30-32.)
73. Graphic Representation of Forces.—A force is said to have three elements. These are (a) its point of application, (b) its direction, and (c) its magnitude. For example, if there is hung upon the hook of a spring balance a weight of 5 lbs., then we have: (a) its point of application on the hook of the balance, (b) its downward direction and (c) its magnitude, or 5 lbs. These three elements may be represented by a line. Thus in Fig. 56a, a line AB is drawn as shown, five units long; A represents the point of application; B, the arrow head, shows the direction; and the length of the line (five units) shows the magnitude of the force.
This is called a graphic representation since it represents by a line the quantity in question. If another weight of 5 lbs. were hung from the first one, the graphic representation of both forces would be as in Fig. 56b. Here the first force is represented by AB as before, BC representing the second force applied. The whole line represents the resultant of the two forces or the result of their combination. If the two weights were hung one at each end of a short stick AC (Fig. 56c), and the latter suspended at its center their combined weight or resultant would of course be applied at the center. The direction would be the same as that of the two weights. The resultant therefore is represented by ON. In order to exactly balance this resultant ON, a force of equal magnitude but opposite in direction must be applied at the point of application of ON, or O. OM then represents a force that will just balance or hold in equilibrium the resultant of the two forces AB and CD. This line OM therefore represents the equilibrant of the weights AB and CD. The resultant of two forces at an angle with each other is formed differently, as in Fig. 57 a. Here two forces AB and AC act at an angle with each other. Lay off at the designated angle the lines AB and AC of such length as will accurately represent the forces. Lay off BD equal to AC and CD equal to AB. The figure ABCD is then a parallelogram. Its diagonal AD represents the resultant of the forces AB and AC acting at the angle BAC. If BAC equals 90 degrees or is a right angle, AD may be computed thus: AB2 + BD2 = AD2. Why?
and AD = √([line]AB2 + [line]BD2).
Fig. 56.—Graphic representation of forces acting along the same or parallel lines.
Fig. 57.—Graphic representation of two forces acting (a) at a right angle, (b) at an acute angle.
This method of determining the resultant by computation may be used when the two forces are at right angles. (In any case, AD may be measured using the same scale that is laid off upon AB and AC, as shown in Fig. 57 b.) The three cases of combining forces just given may be classified as follows: The first is that of two forces acting along the same line in the same or opposite direction, as when two horses are hitched tandem, or in a tug of war. The second is that of two forces acting along parallel lines, in the same direction, as when two horses are hitched side by side or abreast. The third is that of two forces acting at the same point at an angle. It may be represented by the device shown in Fig. 58, consisting of two spring balances suspended from nails at the top of the blackboard at A and B. A cord is attached to both hooks and is passed through a small ring at O from which is suspended a known weight, W. Lines are drawn on the blackboard under the stretched cords, from O toward OA, OB, and OW and distances measured on each from O to correspond to the three forces as read on balance A and B and the weight W. Let a parallelogram be constructed on the lines measured off on OA and OB. Its diagonal drawn from O will be found to be vertical and of the same length as the line measured on OW. The diagonal is the resultant of the two forces and OW is the equilibrant which is equal and opposite to the resultant.
