the less will be the tension. The hammock is an example of this problem. As is well known, there is less danger that the hammock ropes will break if they are more nearly vertical. In the crane shown in Fig. 10 a weight W is supported in its position by forces supplied by the rope AB and the boom AO. These forces act at the point A: the A B F2 W 0 FIG. 10 lines representing the forces form a closed figure. Since the magnitude and direction of W are known, the line W of Fig. 11 is properly drawn as regards direction and length. Starting at D, a line DE is next drawn parallel to the force F1. Next, from C is drawn a line CF parallel to the boom. These lines intersect at a point H. of F1 and F2 unknown. In that case the line W is drawn first, and the problem is solved by finding the triangle which can be constructed by adding two lines of known length. As this is a familiar geometrical problem, the details need not be given here. * The weight of the boom is neglected. Or, the direction and magnitude of the forces F1 and F2 being given, it may be desired to find the weight W which can be supported. The student should be able to explain the necessary steps. 6. Vectors. Lines which by their direction and magnitude represent forces are called vectors. A vector may be used to represent not only force but any quantity which has direction and magnitude. Force, velocity, acceleration, and momentum are vector quantities. Those quantities which have magnitude but do not have direction are called scalar quantities. The number of acres in a farm, the balance in a bank account, the population in the state, a baseball score, and most of the quantities with which we deal in everyday life are scalar quantities. Scalars are added by ordinary arithmetical methods, but vectors are added by geometrical methods. The methods which have just been given for adding forces can be used for any kind of vector quantities. Figs. 9 and 11 are special cases of vector diagrams. 7. Components of a vector. It often happens that not all of an applied force is effective. In Fig. 12 a force F acting in the direction AB is used to pull a barrel up an incline. Only part of this force is effective in rolling the barrel up the incline. In the figure the line AC is drawn parallel to the surface, and the line CB perpendicular to AC. If we add the two vectors AC and CB, the resultant is the vector AB, or F. The vectors AC and CB are called components of the vector AB. The vector AC represents that part of F which produces the motion. The other part, CB, exerts a force perpendicular to the surface of the incline and hence is not effective in producing the desired motion. In Fig. 13 the line AB represents a force used to push a lawn mower in the direction M. The force AB may be broken up into two components AC and CB. The component CB produces the motion in the direction M, but the component AC pushes the mower against the ground and does not assist in moving it. A sled coasting downhill is acted upon by the weight of the sled and its occupants. In Fig. 14 this weight is represented by the vector AB. The right triangle ABC is constructed. The side AC is that component of the weight AB which is effective in moving the sled down the hill. The component CB is a force perpendicular to the ground and contributes nothing to the motion. It is obvious from the figure that if the hill is steeper, the effective component AC is larger. In a form of lever called the knee, or toggle, joint a force applied at F1 produces larger forces, F2 and F3 (see a simplified form in Fig. 15). This lever is often utilized in Any vector may be broken up into two or more components, the only condition being that the sum of these components must equal the original vector. 8. The sailboat. The explanation of how a sailboat can beat into the wind forms an interesting application of the resolution of vectors into components. In Fig. 17 let B represent the direction of the motion of the boat, and W the direction of the wind. The wind acting on the sail can exert a force on the sail only in a direction perpendicular to it. In the figure the vector AD represents the force with which the wind is acting on the sail SS'. The force AD is transmitted through the mast and rigging to the boat. In the lower part of the figure the force AD is shown broken up into the two components AC and CD. CD is the force which tends to make the boat move sidewise. But a sailboat is furnished with a centerboard, or keel, which tends to prevent side drift of the boat. The component AC is the effective one, the force which causes the boat to travel in the direction B. Usually there is some side drift, or leeway, so that the motion is not exactly in the direction B. 9. Forces that produce rotation. So far we have been dealing with forces which tend to produce linear motion of the bodies on which they act. But this is not the only type of motion. For example, in the case represented by Fig. 18 the body A has two forces acting on it which are equal and opposite in direction. B D S W From what has been stated in the preceding sections, one might assume that the sum of these two forces is zero and that no motion would be produced. This is not a correct interpretation of the principles we have had. The two forces will annul each other only when they lie in the same straight line. In the case shown in Fig. 18 a rotation of the body will be produced. It now remains to find out how to treat forces that tend to produce rotation. F A F FIG. 18 10. The moment of force, or torque. In producing rotation about any axis, we learn from experience that a force increases in effectiveness when we apply it farther from the axis. In opening a gate or a door we usually apply a force as far from the hinges as we can. A force applied at a considerable distance from the axis is said to have a greater importance, or moment, than the same force applied nearer the axis. To measure this importance, or moment, the product of the force and the distance of the force from the axis is used. This product is called the moment of force or, frequently, the torque. The precise meaning of the expression "distance of the force from the axis" must be explained. To find this distance a line is drawn from the axis perpendicular to the line of the direction of the force. In Fig. 19 the lengths of the lines a and b are the desired distances. The moment of the 2 force F1 is F1a, and that of F2 is Fab. In the case of the lever shown in Fig. 20 the force F applied at one end will balance a larger force, a weight W, at the other end. The moment of force produced by the force F is Fb, where F F b is the distance to the axis or fulcrum. The greater the distance b, the greater the moment of the force produced by the force F. The moment due to the weight-that is, the product Watends to turn the lever in a direction opposite to that in which the moment Fb tends to turn it. When w these two moments are equal and opposite, the lever is balanced, or in equilibrium. When the two moments of Fig. 20 are equal, we have a b FIG. 20 11. Addition of moments. When two moments of force tend to produce rotation in the same direction, the resultant moment is the sum of the two; when the two act in opposite directions, the resultant is the difference between them. It is customary to call those moments tending to produce rotation in a counterclockwise direction positive, and those tending to produce rotation in a clockwise direction negative. A simple rule is at once formed: The resultant of several moments is their algebraic sum.* This gives not only the magnitude of the resultant but also its direction, * This discussion applies only to forces that lie in the same plane. For the more general case, methods explained in Chapter XI must be used. |