5. A cylindrical hole 16 inches diameter is bored through the centre of a cube 32 inches edge; find the volume remaining. 6. Find the diameter of a spherical body which being thrown into a cylindrical tank whose diameter is 35 inches will make the water rise 2 inches. 7. A cylindrical boiler, inside diameter, 5′ 6′′, and inside length 29' 0". Find the contents in cubic feet and in gallons. Consider two cases, (a) when the ends are spherical, total length 34′ 6′′, (b) ends flat, total length 29' 0". Also in the latter case find the quantity of water in gallons required to fill it to a height of 4 feet. 8. A spherical boiler, 10 feet diameter, is filled to a depth of 7 feet. Find the quantity of water in gallons. 9. A round tower, 120 feet high, is surmounted by a conical top 30 feet high. If the diameter of the tower be 50 feet find the whole exterior surface. 10. If the diameter of a dead-weight safety valve is 3 inches, the pressure of steam 120 lbs. ; find the thickness of each of 12 cast-iron weights, 10 inches diameter, to give the necessary weight. 11. A triangular piece of plate, sides 3 feet 3 inches, 4 feet 9 inches, and 5 feet 2 inches respectively, weighs 2 cwt. Find the weight of a square foot of similar plate. 12. Two buckets have the same depth of 8 inches. One is cylindrical 7 inches diameter, and the other is a conical frustum, the diameter of its ends being 8 inches and 6 inches respectively. Which holds the greatest amount of water, and by how much? 13. A rectangular tank is 7 feet long by 14 feet broad. Find how much the water will rise if a sphere of 9 inches radius be totally submerged in it. 14. A hemispherical dome, 50 feet diameter, is covered with sheet lead weighing 50 lbs. per square foot. Find weight of lead used. 15. A solid metal sphere, 5 inches in diameter, is formed into a tube 13 inches in external diameter and 3 inches in length. the thickness of the tube. Find 16. A cylindrical bar of lead, the diameter of whose section is 23 inches and whose length is 8 inches, is melted down and cast into spherical bullets each ğ inch diameter. How many bullets can be so formed, and what will be the weight of each? Summary. The portion of space included under the three dimensions of length, breadth, and thickness is called the cubical content, the volume, or the solidity of a figure. Total Surface of a Right Prism.-Perimeter of base multiplied by altitude, together with the areas of the two ends. Volume of a Cube.—If a is the length of one side, the volume is a3. Volume of a Right Prism=length x breadth x depth = area of base x altitude, Volume of an Oblique Prism=area of base × altitude. Volume of a Cylinder = area of base × height=πr2 × h. Lateral Surface of a Cylinder = circumference of base × height =2πr x h. Volume of an Oblique Cylinder = area of base × altitude. Volume of a Hollow Cylinder = area of base × altitude=π (R2 – r2) h. Volume of a Pyramid=3(area of base × height). Convex Surface of a Right Pyramid base multiplied by the slant height. = half the perimeter of the Volume of a Cone=3(area of base × height)=3πr2 × h. = Lateral Surface of a Cone (circumference of base × slant height). Lateral Surface of Frustum of a Cone=(sum of circumferences of ends x slant side)=(R+r)l. Surface of a Sphere of Radius r=4πr2. Volume of a Sphere of Radius r=3πr3. Guldin's Theorems.—(1) The area of the surface traced out by the revolution of a curve about an axis in its own plane is equal to the product of the perimeter of the curve and the distance moved through by the centre of gravity of the curve. (2) The volume generated by the revolution of any plane figure about any external axis in its plane, is equal to the product of the generating area and the distance moved through by the centre of gravity of the area. CHAPTER XVIII. LINEAR MOTION. MOMENTUM. FORCE. MASS. WORK. ENERGY Momentum.-When a given mass is in motion, we require to know not only the magnitude of the mass, but also its velocity. The product of the two, or quantity of motion of a body, is known as momentum, thus: Momentum or quantity of motion of a body is the product of its mass and its velocity, or expressed as an equation. Momentum mass x velocity. Unit of Momentum=(unit mass) x (unit velocity). If the units are 1 pound and 1 ft. per second, then the unit of momentum is the momentum of a mass of one pound moving with a velocity of one foot per second. A good example is furnished by the motion of a cannon and a shot. When a shot is fired from a cannon, the momentum generated in both will be the same, but the mass of the cannon is very much greater than that of the shot, hence the velocity or "recoil" of the cannon is much less than that of the shot, and is in the opposite direction. Ex. 1. A cannon weighing 10 tons fires a shot weighing 600 lbs. The shot leaves the gun with a velocity of 1120 feet per second. If the cannon be free to move, find the velocity of recoil. Momentum of cannon = momentum of shot; .. 10 × 2240 × V=600 × 1120; .. V= 600 x 1120 =30 ft. per sec. Hence the cannon would recoil with a velocity of 30 ft. per sec. Ex. 2. The shot in the preceding example, when its velocity is 1000 feet per second, passes through a plate weighing 5 tons, and goes on with a velocity of 200 feet per second. If the plate be free to move, nd the velocity it acquires. As the shot enters the plate with a velocity 1000 ft. per sec. and leaves it with a velocity of 200 ft. per sec., Loss of momentum in passing through plate=600(1000 – 200). But this loss of momentum is acquired by the plate. Hence if V denote the velocity of the plate, we have: Force. Suppose a body to possess a certain amount of momentum, it is impossible for it to alter its state of motion in any manner unless acted upon by something, and that something is termed force. If a force F acting on a mass m during a time t changes the velocity from V1 to v, then or in words, force is rate of change of momentum, and if the time taken be one second we get: Force is momentum per second. Acceleration when uniform is the change in the velocity per unit time. In (1) the change in the velocity is (v — v1), denoting this by f, then F=mf. ..(3) This fundamental equation could also be obtained from (2) by writing for its equivalent f. As Eq. (3) is of the utmost v importance it should be remembered in words thus : Force=mass acceleration, or acceleration force causing motion mass moved When the force F is variable its average or mean value is to be taken. Uniform Force.-Under the action of a constant or uniform force the change in the momentum of a moving body will in any given second be the same as in any other second. Thus, the acceleration produced will be uniform acceleration. Such a uniform force is furnished by the attraction of the earth at any given place upon a mass at or near its surface. This force of attraction, as we have already found, is called the weight of a body. The weight of a body is only another name for the force exerted by gravity on the mass of a body and varies at different places on the earth's surface. It also depends on the distance above sea-level (p. 15). Although its numerical value at various places on the earth's surface is different, at any given place it remains practically constant, and produces a uniform acceleration of g units, hence : If m denotes the mass of a body and W units of force represents the attraction of the earth. Then from Eq. (3) we have or the weight of a body is its mass multiplied by the number of units of acceleration produced by gravity in one second in a body moving freely towards the earth. A weight or force of 1 lb. produces g units of acceleration in unit mass (1 lb.). Hence unit force, called an absolute unit, is part of a pound-roughly, about half an ounce. This is called a poundal. A smaller unit of force is called a dyne, and we have the following definitions: g The Poundal is the name given to that force, which, applied to a mass of one pound for one second generates a velocity of one foot per second. The Dyne is the name given to the force, which, applied to a mass of one gram for one second generates a velocity of one centimetre per second. Gravitation or Engineers' Unit of Force.-What is called the gravitation unit of force is the force equal to 1 lb. Thus, consider the case of a string supporting unit mass (one pound or one gram) the tension in the string is directly equal and opposite to the downward force of gravity, hence the gravitation unit |