(3), that as W' increases, 0 increases. (4), that when bW is very large compared with a W, 0 then ap 2 The first figure gives the position of equilibrium for a given value of W and W' as found from the result (1). The annexed figure gives the position for the same value of W and W' as found from (2). (7). If two forces P and W sustain each other on the arms of a bent lever PCW, whose fulcrum is C, and act in directions PA, WA, which form the sides of an isosceles triangle PAW; shew that if AC be joined, and produced to meet PW in F, P: W:: FW: FP. B W W' The annexed figure represents the system. Since the lever is kept in equilibrium by the forces P, W, and the reaction at C, the resultant of P and W must pass through C; hence AC is its direction, and by the parallelogram of forces, P and W must be proportional to AM and AL respectively, or P: W:: FL: FM. Now, by construction it is evident that PMF and FLW are both isosceles triangles similar to each other, therefore therefore FL: FM:: FW: FP, P: W:: FW: FP. (8). Two equal scale-pans are suspended from the ends of a straight lever, whose arms are as 3 and 4; and an iron bar of 20 lbs. weight is laid upon the scale-pans (just reaching from one to the other): in which of the scale-pans must a weight be placed, and how great must it be, in order to produce equilibrium ? The forces acting here are the weight of the beam and scale-pans, and the weight of the iron bar, both acting in a vertical line passing through the middle point of the lever, the weight required to be placed in the scale-pan nearest the fulcrum, and the resistance of the fulcrum. (9). If the arm of a cork compresser be 18 inches long, and the cork be placed at an inch-and-a-half from the fulcrum, find the pressure produced by a weight of 12 stone suspended from the handle. (10). The whole length of each oar of a boat is 10 feet, and from the hand to the rowlock the distance is 2 feet 6 inches; each of 8 men sitting in the boat pulls his oar with a force of 50 lbs. Supposing the blades of the oars not to move through the water, find the resultant force propelling the boat. Each man pushes back the boat with his feet with a force exactly equal to that which he applies to the handle of the oar, i.e. 50 lbs.; but he presses it forward also by the force which his oar produces upon the rowlock; it is therefore the difference between these forces by which he really pushes the boat forward. (11). What is the ratio between the radii of a wheel and its axle, when a cwt. balances a ton? (12). Two weights P and W are supported on a wheel and axle, P by a string passing round the wheel, W by a moveable pulley whose strings are parallel, and one of them winds on the wheel and off the axle, as P descends: determine the ratio of P to W. This may be solved in two ways: the weight may be considered as applied at any point of the wheel which is vertically above it, in which case the tension of the string must be omitted, for it serves only to keep the weight rigidly attached to the wheel; or again, we may find the tension by considering the weight as supported on a single moveable pulley, and then consider P as kept in equilibrium by the application of this tension both at the wheel and at the axle. (13). In a combination of wheels and axles, in which the circumference of each axle is applied to the circumference of the next wheel, and in which the ratios of the radii of the wheels and axles are 2:1, 4:1, 8:1, and there is equilibrium when the power is to the weight as 1: p; determine the number of wheels. (14). If a man stand in a scale attached to a moveable pulley, and a rope having one end fixed pass under this pulley and then over a fixed pulley, with what force must he hold down the free end in order to support himself, the strings being parallel? (15). P and W represent the power and weight upon the inclined plane: if P's direction lie between the vertical and the normal to the plane, shew that the body must be supported beneath the plane. Assume R to be in the usual direction; its value will be found to be of a negative sign: the interpretation of this is not difficult. (16). If the weight W could be supported on a single moveable pulley by a force P, what must be the inclination of a plane on which the same weight could be supported by the same force applied parallel to the plane? (17). A smooth wedge of given vertical angle is inserted between a horizontal plane and a vertical screw: if a force W be applied to the screw at an arm a, what must be the force acting upon the head of the wedge to preserve equilibrium? (18). A smooth solid body in the shape of a wedge is placed with one side upon a horizontal plane, the other side thus forms an inclined plane; a heavy body is placed upon this inclined plane, and is prevented from sliding down it by a string attached to it, which is fixed to the top of the plain: why does not the pressure of the heavy body perpendicular to the side of the wedge make it slide along the horizontal plane? (19). The arms of a balance are unequal and one of the scales is loaded; a body whose true weight is Plbs. appears to weigh Wlbs. when placed in one scale, and W'lbs. when in the other find the ratio between the arms, and the weight with which the scale is loaded. (20). A shopkeeper has a false balance, and thinks to make his customers' consequent losses and gains balance each other by weighing the goods which he sells alternately in the one scale and in the other: does he succeed? (21). The sliding weight of a Steelyard is 9 lbs. The zero point of graduation is an inch from the fulcrum on the longer arm, and the whole beam will balance about a point 3 inches from the fulcrum on the shorter arm: what is the weight of the beam? (22). In a common Steelyard whose moveable weight is one pound, it is observed that the distance between the point of suspension C and the graduation marked 1lb. is exactly half the distance between C and the point A from which the substance to be weighed is hung; the steelyard alone weighs 6 lbs. Determine the distance of its center of gravity from C. CHAPTER II. SECTION I. ON COUPLES AND SOME PROPOSITIONS RELATING TO FORCES NOT IN ONE PLANE. 65. WE have seen, (Art. 23), that if P and Q, two parallel forces, of which Q is the greater, act in opposite direction at Q-P B points A and B rigidly connected together, their resultant will be a force QP parallel to each of them, and acting at a point Cin AB produced such that If then P do not differ much from Q, Q P is very small compared with Q, and therefore AC is very large compared with AB; that is, the resultant of the two forces is a very small force, and its point of application is at a great distance from A or B. If the difference between P and Q be still less, their resultant is again much smaller, and the distance of its point of application is greater. |