being 3 cwt., and the coefficient of friction between the ladder and wall, and also between the ladder and horizontal pavement which supports its heavier end, being 1 3 1286. A uniform beam AB rests upon a rough horizontal plane AC and against a rough vertical wall CB; show that it will stand when the coefficient of friction, being the same at both ends, is not less than BA-BC 1287. A rough body is supported upon a plane, inclined to the horizon at an angle a, by a string attached to a peg in the surface of the plane; find the least angle which the string can make with the horizontal line drawn through the peg. 1288. Two uniform and equal rough rods, joined at an angle of 120°, are set astride over a rough horizontal cylinder in a plane perpendicular to its axis; find the least angle which one of them can make with the vertical, the coefficient of friction being μ, the length of the rods 2a, the radius of the cylinder r, and the plane of the rods being perpendicular to the axis of the cylinder. 1289. A uniform beam AB, 12 feet long, weighing 100 lbs., is placed upon a vertical post FR, angles FAR and FRA are 45° and 15° respectively, and AF= 4 feet: find the friction at F. 1290. A flat bar weighing 10 lbs. is to be balanced at its middle point upon a horizontal edge, and its centre of gravity is 12 inches from one end and 18 inches from the other: find the greatest force which can be applied obliquely at the lighter end without causing the bar to slip; the coefficient of friction being 1 1291. If a flat uniform bar be supported upon an edge at its middle point, and a weight w be suspended at one end of it, find the greatest angle the direction of any force which will keep the bar horizontal can make with the vertical; μ being the coefficient of friction between the bar and edge, and W the weight of the bar. 1292. A uniform beam length 2a rests against a smooth immoveable hemisphere, of radius r, placed, base downwards, upon a rough horizontal plane upon which one end of the beam is supported. Show that if @ be the greatest inclination of the beam, cot (20+ e) = cos e the coefficient of friction being tan e. 2r sin e, a 1293. Show that, in the wheel and axle, when a force P, acting at the circumference of the wheel, supports a weight the axle, P. (Rp sine) = Q(rpsine) Wp sin e, upon where R, r and p are the radii of the wheel, the axle and their common axis respectively, and is the limiting angle of resistance. 1294. A vertical water-wheel weighing 5 cwt. has a cylindrical bearing of 3 inches radius; find the pressure upon the circumference which will just overcome the statical friction of the axle, the diameter of the wheel being 15 feet, and the coefficient of friction being 2. 1295. A balance having equal and uniform arms 10 inches long, turns upon a cylindrical axle 1 inch in diameter; find by how much a 20 lb. weight in one scale may be exceeded by the weight in the other, before the friction is overcome; the weight of each scale being 1 lb., of each arm 24 lbs., and the coefficient of friction tan 30o. 1296. A uniform lever 2 feet long, weighing 1 lb., is balanced upon an axle whose radius is half an inch, when weights of 10 oz. and 14 oz. are suspended at its extremities, the former weight being about to preponderate; find the length of the arms, if the coefficient of friction be 4. 1297. Show that if P be the force which, acting at the circumference of a horizontal wheel radius R, supported upon the plane end of its cylindrical axle, radius r, just overcomes the fric2ur W tion, P= ; where W is the normal pressure on the bearing. 3R 1298. State Guldin's Theorem for determining the surface of a solid of revolution, and apply it to find the centre of gravity of an arc of a semicircle. 1299. The area of a parabola is two-thirds of that of its circumscribing rectangle, and the volume of a paraboloid is one-half of the volume of a cylinder of the same base and altitude, all the sections of the solid through its axis being equal parabolas. Find the distance of the centre of gravity of any given semiparabola from its axis. 1300. The properties of Guldin being assumed, find the position of the centre of gravity of the arc of a circle in terms of the arc, its chord, and the radius of the circle. 1301. Find, by Guldin's Theorem, the volume generated by the revolution of a parabola about the tangent at its vertex; the curve being limited by its latus rectum. 1302. Find the centre of gravity of the semi-ellipse cut off by the major axis; and thence determine the volume of the solid generated by its revolution round that axis. 1303. A sphere is divided into eight equal parts by three planes perpendicular to one another; find, by integration, the position of the centre of gravity of one of these parts. 1304. Find the centre of gravity of the parabolic spandril ABC; A being the vertex of the parabola, and its axis being parallel to BC. 1305. In a common isosceles trussed roof in which the two struts are parallel to the B opposite rafters, find the tension of the king-post, supposing the weight of the tie-beam to be 1 cwt., of the roof 10 cwt., and of the king-post and of each strut 56 lbs.; and state what would be the effect upon the king-post if the tie-beam were to give way, supposing the parts of the truss to be rigidly connected. 1306. The radius of curvature of a deflected rectangular where E is the modulus of elasticity, b beam being Ecb3 the depth in the plane of curvature, c the breadth and a the length of the beam; show that the amount of deflection at its 4 Pa3 extremity is represented by Ecb 1307. A beam of fir 8 inches square and 10 feet between its horizontal supports, has a 3-inch iron wire securely fastened to its middle point, by means of which increased weights may be suspended until the beam or the wire give way; determine which will first take place, the tabular constant for the breaking strain of fir being 1100 lbs., and the tensile strength of iron wire 36 tons per square inch. 1308. Find the maximum pressure on a vertical revetment wall 20 feet high, the natural slope of the earth being 30°, and a cubic foot of earth weighing 140 lbs.; supposing the friction against the wall to be the same as that of the earth. PRACTICAL MECHANICS. 1309. What allowance per mile must be made in the length of rails upon a railway for a change of temperature of 100°, the modulus of expansion of wrought iron being '00000642 ? 1310. The length of a base line being measured with glass rods was found to be 23702 ft., but the temperature of the rods was observed to be 10° above the standard; what was the true length of the base; the modulus of expansion for glass being ⚫00000431? 1311. Two steel six-foot rods are found to differ from each other by 0035 of an inch, when the difference of their temperatures is 8°.5; at what temperatures will they be accurately the same in length supposing the temperature of the longer one to be kept unaltered at 60°, the modulus being '00000636 ? 1312. The bore of the tube of a thermometer is 014 in. and the quantity of mercury in the thermometer is one-third of a cubic inch; how many inches of scale would there be between 32o and 212°; the modulus of cubic expansion of mercury in glass being '00008696 ? 1313. The weight of an empty thermometer tube is 250 grains, and when filled with mercury it is 1327 grains; find the diameter of the tube when the mercury rises 6.2 inches for a change of 50°. 1314. A steel bar 3 feet long is riveted at one end to a brass bar of the same weight, so that their free ends are towards the same parts. Find the length of the brass bar so that the distance of the centre of gravity of the mass from the free end of the steel bar may be unaltered by change of temperature, the modulus for brass being 00001052. 1315. Supposing the ends of the horizontal bar of an iron railing to be immoveably fixed in two vertical walls; find the thrust exerted upon the walls when the temperature rises 30° F.; the section of the bar being three square inches, the modulus of elasticity 29000000, and of expansion 0000642. 1316. The four iron wire stays of a mast are each one square inch in section and are inclined to the mast at an angle of 30°; find the increase of vertical pressure upon the mast due to a change of 40o F. 1317. The specific gravity of ice (water at 40° F. being the standard) is 91, find its modulus of expansion, supposing the expansion in volume to be three times the linear expansion; and show this supposition to be very nearly correct. |