10. In uniformly accelerated motion, the whole space from rest is proportional to the square of the whole time of motion. 11. The time of falling down an inclined plane of given height is proportional to its length. 12. Twelve pounds' weight is so distributed at the extremities of a cord passing over a pulley, that the more loaded end descends through seven feet in as many hours. What weight is at each end of the cord ? 13. Given the direction, and velocity of projection ; find the range on the horizontal plane, the time of flight, and the greatest height attained by the projectile. 14. Find the length of a pendulum that will oscillate seconds; and of one that will oscillate four times in a second. TRINITY COLLEGE, 1824. 1. EXPLAIN what is meant by the moment of a force, and shew that the moment of the resultant of two forces is equal to the sum of the moments of the components, taken with their proper signs. 2. If three forces, represented in quantity and direction by the three sides of a triangle taken in order, act upon a point, they will be in equilibrium; prove this, by resolving the forces in the direction of rectangular co-ordinates. 3. If a and a', B and B', u and y', be the angles which two forces make with three rectangular axes respectively, and be the angle which they make with each other, then cos. = cos.a cos.cé + cos.ß cos.ß' + cos.y cos.y. 4. A rod of given weight and length, has a moveable weight attached to it, and is placed with one end against a vertical wall, and the other upon a horizontal plane ; find the position of the moveable weight, when a given sustaining force is just sufficient to prevent the rod from sliding when in a given position. 5. If l be the length of the arm of a balance, d the distance of the centres of suspension and gravity, P the load in each scale, and W the weight of the beam, then the sensibility of the balance is 1 , and its stability as d(2P +W). d(2P + W) as a 6. When wheels act by teeth working in one another, the force of one upon the other will remain constant, if the line which is drawn perpendicular to the surfaces of both the teeth at the point of contact, pass continually through the same point of the line which joins the centres of the wheels. 7. Investigate the conditions of equilibrium upon the screw. 8. A uniform beam AB of given weight, is moveable round a hinge at A, and is kept in a given position by a weight P acting by means of a string passing over a pulley at C, and attached to the beam at B; find the weight P, when the points A and C are in the same vertical line. 9. Investigate the differential expressions for the co-ordinates of the centre of gravity of a plane curvilinear surface. 10. When a body, or system of bodies is in equilibrium, its centre of gravity is at the highest or lowest points; prove this principle, and apply it to determine the conditions of equilibrium of two bodies connected by a string, and supported upon a double inclined plane. 11. A rod of given length rests with one end upon the concave surface of an inverted paraboloid, and passes over a point which is in its focus: find the positions in which it rests. 12. A rectangular plane surface ABCD, is moveable round a point at A, and rests against a point at B; from the requisite data, to find the pressure on B. 13. Find the centre of gravity of a frustum of a cone. 14. If A communicate velocity to B, through a number of other bodies which are geometric means between A and B, find the limit to which the velocity of B will continually approach when the number of means is continually increased, the bodies being perfectly elastic. 15. Investigate the following equations, where v is the velocity, t the time, s the space, and f the accelerating force : ds dv (1). (2). f= dt dt 16. Two equilateral triangles are placed with their bases at a given distance from each other upon the same horizontal line, and a V = a non-elastic body falls down the side of the first, along the space between the bases and up the side of the second triangle, the vertex of which it just reaches; given the side of the first triangle, to find that of the second, and likewise the whole time of the motion. 17. P draws up Q by means of a single moveable pulley, when the strings are parallel : find the force accelerating P's descent. 18. The time down a small arc of a circle to the extremity of a vertical diameter is less than the time down the chord. 19. To find the line of quickest descent from a given point to a given inclined plane, and the time of describing it. 20. A body rolls along the curve of an inverted cycloid, descending from the highest point; find the pressure upon the curve at the lowest point. 21. Two bodies, A and B, are projected at the same time from the same point with velocities a and b, one vertically, and the other at an angle of 30°; find the path described by their centre of gravity. 22. If perfectly elastic balls be let fall from the directrix of a parabola whose axis is vertical, and be reflected from the curve, it is required to find, (1). The locus of the vertices of the parabolæ described. (2). The locus of the extreme ranges upon the tangents at the points of incidence of the reflecting curve. TRINITY COLLEGE, 1825. 1. If the sides a, b, c... n of a polygon represent the magnitudes and directions of (n) forces, which keep a point at rest, prove that n?=a? +62 + c + ... -2(ab cos.a, b + ac cos.a, c + bc cos.b,c + ...). 2. (1). Two forces which act in different planes cannot have a resultant. (2). Given the magnitudes and directions of three forces acting in different planes upon a point; find the magnitude and direction of the resultant. a 3. A given weight is supported on two given props which stand at a known distance from each other on a horizontal plane, and which are prevented from sliding, by means of a cord connecting their lower extremities; required the tension of the cord. 4. Any number of parallel forces, and the co-ordinates of their points of application being given, to determine the co-ordinates of the centre of the forces. 5. A given force being applied at a point P within a tetrahedron, it is required to decompose it into four others applied at the angular points A, B, C, D; and thence to prove, that if the lines drawn from these points through P meet the opposite faces in a, b, c, d respectively РЬ Pc Pd + =l. Αα ВЬ Cc Dd 6. Shew that in any polygon, the sum of the squares of the distances of the centre of gravity from the angular points is the least possible. 7. Find the centre of gravity of a given frustum of a paraboloid. 8. Required the solidity of the greatest cone of given diameter which can be supported on a plane whose inclination is 30° ; and find the highest point in the slant side where a weight may be placed without overturning it. 9. A given beam of uniform thickness and density is supported on two planes of known inclination, to find the position in which it will rest. Pa 10. (1). When a system is in equilibrium, the centre of gravity is the highest or the lowest possible: hence, (2). Determine the position in which two given beams of uniform density will rest, when they are placed on a horizontal plane with their lower extremities opposed to each other, and their other extremities supported on two parallel vertical planes. 11. In toothed wheels, the moment of P about the centre of the first wheel is to the moment of W about the centre of the second wheel, as the perpendiculars from the centres of the wheels upon the line of direction of their mutual action. 12. A homogeneous elliptical spheroid rests on its smaller end, in a concave hemisphere; to find what the radius of the hemisphere must be that the equilibrium may be stable. 13. (1). Does the force of friction observe any known law? Is the quantity of friction affected by an increase of velocity, or by the extent of the surfaces that are in contact ? Explain how friction may be measured, and mention the chief contrivances that are employed to diminish it. (2). Prove that the friction of a rope, which is wound round a cylinder, increases in geometrical progression, while the number of turns increases in arithmetical progression. 14. Two bodies P and Q are connected by a string passing over a fixed pulley; P descends vertically, and draws Q along the horizontal plane ; find the space described and the velocity acquired by P in t". 15. If a body be projected vertically upwards or downwards with a velocity (u), and be acted upon by a constant force (f ), prove that s= tu F ift?. 16. The axis of an equilateral hyperbola being vertical, prove that the times of descent from the extremities of the axis to any point in the curve are equal. 17. (1). The position of a ball on a triangular billiard table being given; it is required to shew that there are three directions, in any one of which, if the ball be struck, it will pursue continually the same path after being twice reflected from each of the sides. (2). Find the position of the ball, and the direction in which it must be struck so that it may pursue continually the same path after being once reflected from each side. 18. A shell being discharged at a given angle, the sound of its explosion was heard at the mortar, t" after the discharge ; required the horizontal range. 19. Let A, P be two points in the same vertical plane, join A, P meeting in Q an inverted cycloid AQB, whose base AB is part of the horizontal line AX; join QB, parallel to which draw the line PC meeting AX in C: then AC is the base of an inverted |