and prove that the portion of the tangent intercepted between the coordinate axes is of constant length. 6. If p be the perpendicular from the origin on the tangent to a curve at a point whose radius vector is r, prove that the radius of curvature at the Apply this to determine the length of the radius of curvature at any point on a parabola. 7. Trace the curve r=a cos 40+b sin 40, and find its entire area. 8. Determine the asymptotes to the curve x3+3x2y + 2xy2+ zaxy+2ay2 − 2a3=0. 9. Integrate the expressions x2 dx sin3 x cos x dx, x2 log x dx, x6 - I IO. II. Find the values of any two of the definite integrals Prove that the whole area of a cycloid is three times that of its generating circle. 12. Rectify either of the curves (1) ay2=x3, (2) x3+y3=a3. 1. VIII. STATICS. [Great importance will be attached to accuracy.] Two forces are given in magnitude, but may make any angle with each other. How should they be placed, so as to give (1) the greatest, (2) the least possible resultant? Prove your statements. 2. What is meant by a force resolved in a given direction? Show how a force is to be resolved so that its component along a given direction shall have a given value. Give a geome 3. A force is given in magnitude and line of action. trical construction for resolving it into two other forces which shall be equal to one another and shall pass respectively through two fixed points. 4. Prove that any two couples of equal moments and opposite senses balance each other. (It may be assumed that the same couple can be transferred in its own plane in any manner without changing its effect.) 5. Three parallel forces act on a horizontal bar. Each is = 1 lb.; the right-hand one acts vertically upwards, the two others vertically downwards at distances 2 ft. and 3 ft. respectively from the first; draw their resultant, and state exactly its magnitude and position. 6. Draw any system of pulleys by which a weight of 1 lb. can be made to support a weight of 3 lbs., neglecting friction and the weights of the pulleys. Show that, whatever may be your system, the smaller weight will descend through 3 ft. in raising the other through 1 ft. 7. A heavy circular disc is kept at rest on a rough inclined plane by a string parallel to the plane and touching the circle. Show that the disc will slip on the plane if the coefficient of friction is less than tani, where i=slope of plane. 8. Two equal weights each = 112 lbs. are joined by a string which is laid over two pulleys A, B in the same horizontal line. If a small weight, say I lb., is attached to the string half way between A and B, find in inches the depth to which it descends below the level of AB: supposing AB= 10 ft. What would happen if the weight were attached at any other point of the string? 9. If a portion m of any mass M is moved to any new position, show that the centre of gravity of the entire mass is thereby moved in a direction parallel to the displacement of the centre of gravity of m, and over a D, where D = above distance between the two positions of the centre of gravity of m. A triangular piece of paper is folded across the line bisecting two sides, the vertex being thus brought to lie on the base. Find the centre of gravity of the paper in this position. IO. Three equal particles are placed anywhere on the three sides of a triangle. If they are moved along those sides, in the same sense, and over three spaces which are proportional respectively to the sides, show that the centre of gravity of the particles remains at rest. II. Assuming the principle of virtual velocities, deduce the relation between the power and weight on the inclined plane; the power being either (1) parallel to the plane or (2) horizontal. 12. Explain what is meant by stable and unstable equilibrium, and give an instance of each. IX. DYNAMICS. [The measure of the acceleration of gravity may be taken to be 32 when a foot and a second are units of length and time. Great importance will be attached to accuracy.] 1. Explain how velocity is measured, and if 22 be the measure of a velocity when a foot and a second are units of length and time, find its measure when a mile and an hour are the units. A man, 6 feet in height, walks with the velocity of 3 miles an hour, in a straight line along a road, on one side of which there is a lamp-post, the light of the lamp being 9 feet above the ground. Find the velocity of the end of his shadow on the ground. 2. A ship, which is sailing due north at the rate of 3 miles an hour, observes another ship exactly east of it, which is sailing due east at the rate of 4 miles an hour. Find the rate at which each ship is increasing its distance from the other, and determine graphically the direction of motion of each relative to the other. 3. Define acceleration, and explain how it is measured. If a point, starting with no velocity, moves with a constant acceleration f in the direction of its motion, and passes over the space s in the time t, and if v is its velocity at the end of that time prove that 2s=ft2, and v2 = 2fs. If the point starts with the velocity u, and moving with the constant acceleration f, passes over the spaces, where is the error in the statement that the final velocity is u+ √2fs? 4. Enunciate and explain Newton's second law of motion. A string passing over a smooth pulley supports two scale-pans at its ends, the weight of each scale-pan being equal to the weight of one ounce. If a two-ounce weight be placed in one scale-pan, and a four-ounce weight in the other, find the acceleration of the system, the tension of the string, and the pressures between the weights and the scale-pans. 5. Prove that the time of descent of a heavy particle down any chord of a vertical circle, starting from the highest point of the circle, is the same. Find the line of quickest descent from a given point to a given circle in the same vertical plane. 6. Prove that the path of a projectile is a parabola, and find the greatest possible range on the horizontal plane through the point of projection for a given velocity of projection. Show that, for any range short of the greatest, if the velocity of projection is given, there are two directions of projection which are equally inclined to the direction giving the greatest range. 7. If a point moves uniformly, with velocity v, in a circle of radius r, find the direction and the measure of its acceleration. If a particle of mass m moves in the same circle with the same velocity, find the direction and magnitude of the resultant of the forces which are in action on the particle. A heavy particle, which is suspended from a fixed point by a string one yard in length, is raised until the string (which is kept tight) is inclined 60° to the vertical, and is then projected horizontally, in the direction perpendicular to the vertical plane through the string; find the velocity of projection that the particle may move in a horizontal scale. 8. Find the velocity acquired by a heavy particle in sliding down a smooth curve. A heavy particle is placed very near the highest point of a smooth sphere; find where it runs off the sphere, and prove that the latus rectum of the parabola, which it then describes, is eight twenty-sevenths of the diameter of the sphere. 9. Define the potential energy and the kinetic energy of a system, and enunciate the principle of energy. A straight rod ACB, without weight, has two particles of equal weight fastened to it, one at the end B, and the other at the middle point C, and the rod can swing about the end A. If it be held horizontally, and then allowed to swing, prove that the greatest velocity acquired by the end B will be the same as the velocity acquired by a particle falling freely through a height equal to six-fifths of the length of the rod. IO. A cannon standing on a smooth horizontal plane is pointed horizontally and loaded with a ball, the mass of which is a given fraction of the mass of the gun and its carriage. Having given the velocity with which the ball leaves the gun when it is fired, find the velocity of recoil of the gun. If the charge of powder be quadrupled, what will be the effect on the velocities? |