84. If a uniform chain be suspended from two piers, the points of suspension being in the same horizontal line; shew that when the chain is nearly horizontal, the tension is nearly 32 equal to the weight of a length of the same chain, 4(8 - b) length of the chain, and b = the distance between the points of suspension. Shew also that such a length may be given to the chain as to render the tension at either pier a minimum; and investigate an equation for determining the minimum tension. where s = 85. Three uniform beams AB, BC, CD, of the same thickness, and of lengths l, 21, 1 respectively, are connected by hinges at B and C, and rest on a perfectly smooth sphere, the radius of which = 21, so that the middle point of BC, and the extremities of A, D are in contact with the sphere; shew that 91 the pressure at the middle point of BC = of the weight of 100 the beams. 1832 86. Graduate the common steelyard. 88. If two weights acting perpendicularly on a straight lever on opposite sides of the fulcrum balance each other, they are to one another inversely as their distances from the fulcrum. 89. Find the force requisite to draw a carriage wheel over an obstacle, supposing the weight of the carriage collected at the axis of the wheel. 90. Find the relation between P and W in equilibrium, on a system of pulleys where each string is attached to the weight, supposing the weights of the pulleys to be equal. 91. State generally the principle of virtual velocities. If any point of the system in equilibrio press against a resisting surface, prove that the virtual velocity of the normal force is nothing. Apply the principle to the following problem. Two bodies P and Q are connected by a rigid rod, one of which rests on a given plane; determine the nature of the curve on which the other must rest that they may be in equilibrio in all do 4 93. Find the forces parallel to the axes of an ellipsoid that keep a particle at rest on every point of its surface; prove that the resultant of all the forces varies inversely as the perpendicular from the centre on the tangent plane. 94. Explain the action of toothed wheels on each other, and find the ratio of P to Win the case of equilibrium. 95. Find the magnitude and point of application of the resultant of any number of parallel forces acting on a rigid body. Explain the meaning of the result wlien there are only two equal and opposite forces. What is their moment round any point in the plane in which they act ? 96. A uniform catenary of given length is suspended from two given points at the same height, and is nearly horizontal; in consequence of an expansion of its materials the vertex of the catenary is observed to have descended through a small given altitude; find the increase of the length of the catenary, supposing its expansion to have been uniform throughout. 97. CA and CB are the arms of a uniform bent lever; determine the distance of its centre of gravity from C, having given the lengths of the arms and the angle ACB. 98. A uniform beam rests with one end against a smooth vertical wall, and with the other on a horizontal plane, the 1 friction on which is - th of the pressure; determine the inclination of the beam to the horizon when just supported by the friction. n 99. Find the centre of gravity of a triangular pyramid, and prove that its distance from the base is ath of the altitude. 100. Find the proportion of the power to the weight when there is equilibrium on the inclined plane, the power acting in any given direction, 1833 . are the 101. If any number of forces P, Q, . . . and P, Q',.. acting upon the arms of a lever to turn it in opposite ways round a fixed point C, be such that P.CM + Q.CN +... = P'. CM' + Q'. CN' +... where CM, CN, perpendiculars from C on the directions of P, Q, .. there will be an equilibrium. 102. Having given the coordinates of any number of bodies, considered as points in the same plane; determine those of their centre of gravity. 103. Find the ratio of the power to the weight on a system of pullies, in which each pulley hangs by a separate string, and all the strings are parallel. 104. Having given that the resultant of two forces, applied at a point, is in the direction of the diagonal of the parallelogram, whose sides represent the forces in magnitude and direction; shew that it is represented in magnitude by the diagonal. 105. The pressure on the fulcrum of a lever, acted on by any number of forces in the same plane, is equal to the resultant of all the forces, supposing them applied at that point, retaining their directions. 106. An ellipsoid rests on a horizontal plane on the extremity of its mean axis; shew how to estimate the stability with regard to a slight displacement in any direction. Define the direction which distinguishes between stable and unstable equilibrium. 107. Apply the principle of virtual velocities to shew, that when three forces, acting perpendicularly upon the sides of a scalene wedge, keep each other in equilibrio, they are proportional to those sides. 108. If a rope applied to the arc of any curve be drawn by two forces acting at its extremities, and one of them be on the point of preponderating; prove that it is greater than the other in the ratio of ena : 1, where n is the ratio of friction to pressure at every point of the arc, a the angle between the normals at the points where the rope leaves the curve, and ε = 2.7182818. 109. Let p,p' be two forces into which a given systein acting upon a rigid body may be resolved; a, 0, the least distance and inclination of their directions; prove that pp'a sin 0 is invariable; also, if the same system of forces be resolved into a single resultant force and a single couple; prove that the moment of the couple multiplied by the sine of the angle which its plane makes with the resultant, is invariable. 110. In a plane triangle, if the line joining the centre of the circumscribed circle, and the point of intersection of the perpendiculars be trisected, the point of division which is nearest to the centre of the circumscribed circle is the centre of gravity of the area of the triangle. 111. The centre of gravity of three weights a.(w – a), b.(w – B)?, c.(w – 7)?, whatever be the value of w, will be situated in a line of the second order to which the lines joining the centres of gravity of the weights are tangents. 112. Required the equation to the common catenary sus- 1834 pended from two points in the same horizontal line. Shew how the constant may be determined when the length of the chain and the distance between the points of suspension are given. 113. In a system of parallel forces, having given the distances of the points of application from one another, and from a fixed point, find the distance of the centre of the system from that point. Apply the result to find the distance of the centre of gravity of a triangular pyramid from one of the angular points ; assuming that it coincides with the centre of gravity of four equal bodies placed in the angular points. 114. Find the necessary equations of equilibrium of a rigid body acted on by any forces, and state the modifications of those equations, when, instead of being free, the body has one point immoveable, or two points immoveable. 115. Define the terms couple and axis of a couple ; if the magnitudes of the moments and directions of the axes of two couples be represented by adjacent sides of a parallelogram, 1834 prove that its diagonal will represent in moment and axis the resultant couple. 116. Explain the effect of friction in supporting an arch. How is the true theory of the arch connected with the theory of roofs ? 117. If A, B, C represent the moments of a force round each of three rectangular axes which meet in a point, and a, ß, Υ be the angles which a straight line through the point of intersection makes with each axis, the moment of the force round this line is A cos a + B cos ß + C cos y. 118. Enunciate Guldinus' properties of the centre of gravity, and apply one of them to find the distance of the centre of gravity of the area of a seini-parabola from its axis. 119. State the principle of virtual velocities, and thence deduce the relation of the power to the resistance on the screw. 120. If any number of forces P, Q, . . . and P', Q', . in the same plane acting upon the arms of a lever to turn it in opposite ways round a fixed point C, be such that P. CM + Q.CN + ..=P.CM' + Q'.CN' + where CM, CN, ... are the perpendiculars from C on the directions of P, Q, . . ., there will be an equilibrium. State the axioms and propositions which you assume in this proof. 121. If three forces act on a point in the directions of the sides of a triangle taken in order, and keep it at rest, they are represented in magnitude by the sides of the triangle. Is the same true of the sides of a polygon? 122. When P begins to move vertically from the state in which it balances W on the single moveable pulley with strings not parallel, P's actual velocity : W's actual velocity ::W:P. Does the proposition thus enunciated hold for all the mechanical powers ? 123. In the combination of levers used in the Stanhope printing-press, find the relation between the pressure exerted and the force applied ; and make the advantages of such a combination appear. . |