124. If a heavy cord passing over two pullies, fixed in a horizontal line, be kept at rest by equal weights attached to its extremities, prove that no possible increase of the weights can stretch it so as to become horizontal. 125. A number of forces in one plane act upon a point in given directions; determine the magnitude and direction of their resultant. How may both be expressed by one algebraical formula ? 126. Define the centre of gravity, and shew how to find its position for two bodies considered as points. If a spherical surface be cut by two parallel planes, prove that the centre of gravity of the intercepted portion bisects the line joining the centres of the circular ends. 127. If a lever, kept at rest by weights P, Q, suspended from its arms a, b, so that they make angles a, ß, with the horizon, be turned about its fulcrum through an angle 20, prove that the vertical spaces described by P and Q, are to one another as a cos(a + ): b cos (B - 0); and thence deduce the equation of virtual velocities. 128. If S and D represent respectively the semi-sum and semi-difference of the greatest and least angles, which the direction of a power supporting a weight on a rough inclined plane may make with the plane, and o be the least elevation of the plane when a body would slide down it; prove that the cosine of the angle, at which the same power being inclined to a smooth plane of the same elevation would support the same weight, cos (D + 0). cos 129. Explain the weighing machine for turnpike roads. 130. If a right-angled triangle be supported in a horizontal position by vertical threads fastened to its, angular points, each of which can just bear an additional tension of 1 lb.; determine within what portion of the area a weight less than 3 lbs. may be placed without destroying the equilibrium. 131. Two forces balance each other on a lever moveable about a cylindrical axis; shew that the centre of the axis must cos S M lie in the direction of their resultant. Find also how far it may be removed from this line, when friction is considered, before motion will ensue. 132. Find the form of an elastic lamina of uniform breadth and thickness, fixed at one end, and acted upon at the other by a given force, mentioning the hypotheses on which the investigation proceeds. 133. Find the conditions of equilibrium of any number of forces acting in the same plane upon a rigid body, and apply them to determine the position of a beam resting upon a point with one end against a vertical plane. 134. A combination of mechanical powers consists of a cylinder (turned by a winch) on which is the thread of a screw working in the teeth of a wheel; and round the axle of the wheel passes a cord, drawing a weight up an inclined plane to which its direction is parallel. Compare the force turning the winch with the weight drawn up the plane. 135. If the inclination to the horizon of a plane on which a body is placed, be slowly increased till the body begins to move; shew that its tangent, at the instant motion begins, expresses the ratio of friction to pressure. 136. Explain the construction and graduation of the common steelyard. On what account is the common balance preferable in determining small weights? 137. Find the relation of P to W in the isosceles wedge. 138. Shew in what case a body will remain at rest when placed on a horizontal plane. 139. If three forces keep a point in equilibrium, and three lines be drawn making with the directions of the forces three equal angles towards the same parts; these three lines will form a triangle whose sides will represent the three forces respectively. 140. By what experiments is it proved that the friction between the same substances depends only upon the pressure ? Shew how the coefficient of friction may be practically determined. 1835 141. If in a system consisting of any number of particles a point be taken, and if each particle be multiplied by the square of its distance from the point, the sum of these products will be the least when the point is the centre of gravity. 142. Find the centre of gravity of the surface of an equilateral spherical triangle. 143. The centre of gravity of a system of bodies, placed in any given position, and acted upon only by the force of gravity and the reactions of the surfaces upon which the particles move, will continually descend until it becomes the lowest possible. 144. Explain the action of an oar when used in rowing; and determine the effect produced, having given the distances of the fulcrum and the hand of the rower from the side of the boat. 145. If any number of forces act in the same plane on a rigid body, determine the condition that they may have a single resultant. 146. A roof ACB, consisting of beams which form an isosceles triangle with its base AB horizontal, supports a given weight at C; find the horizontal force at A. Why must a pointed arch carry a heavy weight at its vertex ? 147. How is a notion of force acquired ? Give a definition 1836 of it. What is a resultant? Assuming the resultant of two forces, determine that of any number of forces acting on a point. 148. Find the distance of the centre of gravity of any number of bodies, in the same straight line, from a point lying between two of the bodies. 149. Determine the relation of P to Win a system of pullies, where the strings are parallel and each attached to the weight; the weights of the pullies being taken into account. 150. If two forces, acting perpendicularly on a straight lever on the same side of the fulcrum, are inversely as their distances from the fulcrum, they will balance each other. A uniform heavy rod, at a given point of which a given weight is attached, is sustained at one end; determine its length when the force, which applied at the other end will keep it horizontal, is a minimum. 151. Explain and illustrate the proposition “ in any machine, what is gained in power is lost in time, and conversely.” Point out the principal advantages gained by the use of machines. 152. A body, the lower surface of which is spherical, rests upon a sphere; find in what case the equilibrium is stable. 153. Three forces act on a point in directions respectively perpendicular to three rectangular coordinate planes, and each varying as the coordinate to which it is parallel ; shew. that there are two planes, in either of which if the point be situated the resolved part of the whole force, which is parallel to the plane, tends to the origin and varies as the distance of the point from it. 154. If the vertical angle of a right cone of circular base be greater than sin-'1, the frustum cut off by any plane will be supported with its base on a horizontal plane. If the vertical angle be less than sin-1, determine the limits for the inclination of the cutting plane to the axis that the frustum may stand. 155. The extremities of a given flexible and uniform heavy chain are attached to unequal arms of a straight lever; investigate an equation to find the position of equilibrium of the lever, neglecting its weight. 156. Find the resultant of two parallel forces acting on a rigid body, and thence that of any number. 157. State briefly the results of experiments on friction. Investigate the condition of equilibrium when two forces act at the extremities of a rope coiled about a rough cylindrical axle; and thence explain the effect of coiling the rope about the axle in working a capstan. 158. State and prove Guldin's property of the centre of gravity, by which the volume of a solid of revolution may be determined. Apply it to find that of the frustum of a right cone in terms of its altitude and the radii of its ends. 159. A rigid body is acted on by a system of forces which have not a single resultant; shew that they may be reduced to a single force and a couple whose plane is perpendicular to the direction of the force. Find the equations of the line in which the force acts, referred to any origin and rectangular axes, and the moment of the couple. 160. In the single moveable pulley, the strings not parallel, shew that P. P's velocity = W.W's velocity. = W.W’s velocity. Determine whether the equilibrium is stable or unstable. |