6. If a straight line be divided into any two parts, the rectangle contained by the whole and one of its parts, is equal to the rectangle contained by the two parts, together with the square on the aforesaid part. Shew that the rectangle contained by the two parts is greatest when the given line is bisected. 7. 8. To describe a square that shall be equal to a given rectilineal figure. 9. If one circle touch another internally in any point, the straight line which joins their centres being produced shall pass through that point. Describe a circle of given radius which shall touch internally a given circle in a given point: the given radius being less than that of the given circle. 10. Equal straight lines in a circle are equally distant from the centre. 11. The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is, upon the same part of the circumference. 12. If the vertical angle of a triangle be divided by a straight line which also cuts the base, and the segments of the base have the same ratio which the other sides of the triangle have to one another, the vertical angle of the triangle shall be divided into two equal angles. MECHANICS AND HYDROSTATICS. FRIDAY, January 11, 1856. 9 to 12. FIRST DIVISION.—(A.) 1. IF two forces acting perpendicularly on a straight lever in opposite directions and on the same side of the fulcrum balance each other, they are inversely as their distances from the fulcrum; and the pressure on the fulcrum is equal to the difference of the forces. If the distances from the fulcrum of two forces acting in opposite directions and keeping it at rest be three and five feet respectively, and the pressure on the fulcrum be three pounds, what are the magnitudes of the forces? 2. If the adjacent sides of a parallelogram represent the component forces in direction and magnitude, the diagonal which passes through the intersection of these sides will represent the resultant force in direction and magnitude. If the resultant force be represented in direction and magnitude by the diameter of a circle, and one of the component forces by a given chord passing through one extremity of that diameter, give a geometrical construction for representing the other component. 3. In a system of pullies in which each pully hangs by a separate string and the strings are parallel, there is equilibrium when P: W:: 1: that power of 2 whose index is the number of moveable pullies. If P be equal to six pounds and W to forty-eight pounds, how many moveable pullies will there be? 4. The weight (W) being on an inclined plane, and the force (P) acting parallel to the plane, there is equilibrium when P: W:: the height of the plane: its length. W If P be equal to what will be the inclination of the plane? 5. Define velocity, and shew that if P and W balance each other in the manner described in the preceding question, and the whole be put in motion, P: W:: W's velocity : P's velocity. If the number of moveable pullies be four (the system being that in which each hangs by a separate string) and P's velocity be 32 feet in a second, what will be W's velocity? 6. When a body is suspended from a point, it will rest with its centre of gravity in the vertical line passing through the point of suspension. A right-angled triangle is suspended by its right angle, and the inclination of the hypothenuse to the horizon is forty degrees, find the acute angles of the triangle. 7. The pressure upon any particle of a fluid of uniform density is proportional to its depth below the surface of the fluid. Find the height of a column, standing in water 30 feet deep, when the pressure at the bottom is to the pressure at the top as 3 to 2. 8. Explain the hydrostatic paradox. 9. When a body of uniform density floats on a fluid, the part immersed : the whole body:: the specific gravity of the body: the specific gravity of the fluid. Find the specific gravity of a material such that a cylinder formed of it four inches long floats in water with three inches immersed. 10. Describe the common hydrometer, and shew how to compare the specific gravities of two fluids by means of it. 11. Having given the number of degrees on Fahrenheit's thermometer, find the corresponding number on the Centigrade thermometer. What is the temperature when the number of degrees on the Centigrade thermometer is as much below zero, as that on Fahrenheit's is above? FIRST DIVISION.-(B.) 1. Ir two weights acting perpendicularly on a straight lever on opposite sides of the fulcrum balance each other, they are inversely as their distances from the fulcrum, and the pressure on the fulcrum is equal to their sum. If the distances from the fulcrum of two forces acting in the same direction and keeping it at rest be 4 and 6 feet respectively, and the pressure on the fulcrum be 25 pounds, what are the magnitudes of the forces? 2. If three forces represented in magnitude and direction by the sides of a triangle taken in order, act on a point, they will keep it at rest. Two forces whose magnitudes are 12 and 5 pounds respectively, act at right angles to each other on a given point; what is the magnitude of the force which will keep the point at rest? 3. Describe the Wheel and Axle; and prove that there is equilibrium when the power is to the weight as the radius of the axle to the radius of the wheel. 4. In a system of pullies in which the same string passes round all the pullies, and the parts of it between the pullies are parallel, there is equilibrium when the power is to the weight as 1 to the number of strings at the lower block. What will be the magnitude of the weight, when it exceeds the power by 40 pounds, and there are six strings at the lower block? 5. Define velocity, and assuming that the arcs which subtend equal angles at the centres of two circles are as the radii of the circles, shew that if P and W balance each other on the wheel and axle, and the whole be put in motion, P: W:: W's velocity: P's velocity. If W exceed P by nine pounds, and P's velocity exceeds W's by six feet per second, the sum of P and W being eleven pounds, find W's velocity. 6. When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line drawn from its centre of gravity falls within or without its base. A circular table is supported by three legs meeting it in its circumference and two of the angles of the triangle formed by its feet are 30 degrees and 45 degrees respectively, will the table stand or fall? State your reasons. 7. The surface of every fluid at rest is horizontal. Why is the mast-head of a ship at sea seen before the hull? 8. If a body floats on a fluid, it displaces as much of the fluid as is equal to the weight of the body; and it presses downwards, and is pressed upwards with a force equal to the weight of the fluid displaced. If a cubic foot of water weigh 1000 ounces, and a cube whose edge is 18 inches, weigh 2250 ounces, how far will a cylinder whose length is 3 inches, and formed of the same material as the cube, sink in water? 9. Define specific gravity, and prove that if M be the magnitude of a body, Sits specific gravity, and Wits weight, W=MS. Find the specific gravity of the material mentioned in the above question, and if it be united with half its bulk of a material whose specific gravity is , find the specific gravity of the compound. 10. Describe the hydrostatic balance, and shew how to find the specific gravity of a body by means of it; first when its specific gravity is greater than that of the fluid in which it is weighed, and secondly when it is less. 11. Describe the construction of the common air-pump, and its operation. 1. DEFINE force. Shew that forces can be properly represented by straight lines. Apply forces of 1, 2, 5 and 7 lbs. respectively to a point so as to give the smallest possible resultant; the forces all acting in the same straight line. 2. Assuming that the effect of a force to turn a lever round its fulcrum varies as the force, when the arm is constant, and as the arm when the force is constant, shew that, when two forces balance on a lever, they are inversely proportional to the arms of the lever. If two forces balance on a straight lever, when their directions are at right angles to the arms, they will balance when their directions make any equal angles with the arms. 3. Define resultant force. If the adjacent sides of a parallelogram represent two forces in magnitude and direction, prove that the diagonal which passes through the intersection of the sides will represent the resultant force in direction. Given the direction and magnitude of the resultant force, determine the directions of the component forces, when they are each equal to the resultant. 4. There is equilibrium on the wheel and axle, when the power is to the weight as the radius of the axle is to the radius of the wheel. Is there any advantage in having the rope which passes round the wheel thicker than that which passes round the axle ? 5. In the system of pullies in which each pully hangs by a separate string, and the strings are parallel, there is equilibrium when P: W:: 1: that power of 2 whose index is the number of moveable pullies. 6. Find the centre of gravity of two heavy points. Two equal particles are placed on two opposite sides of a parallelogram, shew that their centre of gravity will remain in the same position, if they move along the sides so as always to be equidistant from opposite angles. 7. The pressure upon any particle of a fluid of uniform density is proportional to its depth below the surface of the fluid. In two uniform fluids the pressures are the same at the depths of 3 and 4 inches respectively, compare the pressures at the depths of 7 and 8 inches respectively. 8. When a body is immersed in a fluid, the weight lost: whole weight of the body: the specific gravity of the fluid: the specific gravity of the body. A body whose specific gravity is 2.7 and weight in vacuo 3 lbs., when immersed in a fluid weighs 2lbs.; find the specific gravity of the fluid. 9. Describe the common hydrometer, and shew how to compare the specific gravities of two fluids by means of it. 10. Shew how to graduate a common thermometer. What would be the inconvenience of having the bore of the thermometer large? 11. Describe the construction of the forcing pump and its operation. SECOND DIVISION.-(B.) 1. DEFINE weight. How is force measured in Statics? If a force act upon a body, what must be known concerning it that its effect may be wholly determined? 2. Assuming that the effect of a force to turn a lever round its fulcrum varies as the force when the arm is constant, and as the arm when the force is constant, shew that, when two forces balance on a lever, they are inversely proportional to their distances from the fulcrum. Forces in the ratio of 3: 2 and on opposite sides of the fulcrum balance each other, find the position of the fulcrum, the length of the lever being 10 inches. 3. Define Component forces. If the adjacent sides of a parallelogram represent two forces in magnitude and direction, then assuming that the diagonal passing through the intersection of these sides represents their resultant in direction, prove that it will represeut it in magnitude. If the two forces are each equal to 6 lbs., and their directions inclined at of a right angle, shew that their resultant equals 6√√3lbs. 4. The weight (W) being on an inclined plane, and the force (P) acting parallel to the plane, there is equilibrium when P: W: the height of the plane its length. If the inclination of the plane be half a right angle, find P in terms of W. 5. In a system of pullies in which the same string passes round any number of pullies, and the parts of it between the pullies are parallel, there is equilibrium when P: W:: 1: the number of strings at the lower block. 6. Find the centre of gravity of a straight line. A wire is bent so as to form three sides of a square, find its centre of gravity. |