If a particle moves in a straight line so that its velocity is proportional to the square of its distance from a fixed point, shew that the force on it varies as the cube of the distance from the same point. 6. A particle moves in a straight line under an attractive force varying as the distance from a fixed point. Investigate the motion. A particle is acted on by two equal forces constant in magnitude, but whose directions revolve at equal constant rates in opposite directions. Find the motion. 7. State Kepler's Laws, and deduce the law of attraction of the sun on a planet. 8. Prove that in a fluid in equilibrium the rate of increase of pressure in any direction is equal to the product of the density and the resolved part of the force per unit mass in that direction. 9. Define carefully the term metacentre, and prove the formula for its height above the centre of gravity of the liquid displaced ffx2dxdy HM= V ¦ I MIXED MATHEMATICS.-PART III. The Board of Examiners. 1. Starting from the definition of a couple prove that its effect on a rigid body depends only on its moment and the direction of the normal to its plane. 2. Investigate the form taken by a uniform heavy string suspended from two points showing that referred to proper coordinates its equation is 3. Demonstrate Stokes' Theorem that the integral of normal attraction over a closed surface is 4πη where m is the mass inside the surface. Deduce the law of attraction outside a solid sphere. 4. Find expressions for the components of acceleration of a particle moving in a plane in polar coordinates. 5. Find the motion of a particle in a straight line under a central force varying as the inverse square of the distance, and show that the time a3 from rest at distance a to the centre is a 8μ is the astronomical mass of the attracting centre. where u 6. Investigate the motion of a simple pendulum, showing that the time of a complete oscillation is neglecting the fourth power of the amplitude a, where I is the length of the pendulum. 7. Given the moments and products of inertia of a mass about three rectangular axes at a point, investigate the expression for the moment about any other line through the point. How would you obtain the moment about a line not through the point, the centre of gravity being supposed known? 8. Use D'Alembert's principle to obtain the equations of motion of a rigid body in two dimensions Mk20 = L. Mä = X Mÿ = Y 9. Prove the equation of energy Σ M (x2 + y2 + k2 02) = 2 Σƒ (X dx + Y dy + Z dz) + C for motion in two dimensions, and apply it to show that in the case of a sphere rolling down a perfectly rough inclined plane the motion of the centre is the same as if the plane were smooth, and the intensity of gravity decreased in the ratio 5: 7. PURE MATHEMATICS.-PART I. The Board of Examiners. 1. The opposite angles of any quadrilateral inscribed in a circle are together equal to two right angles. Show that a circle can always be drawn through the angular points of a quadrilateral having two opposite angles together equal to two right angles. 2. If the vertical angle of a triangle be bisected by a straight line which cuts the base, the segments of the base shall have to one another the same ratio as the remaining sides of the triangle. An isosceles triangle ABC has the angles B, C, each double of A, and the angle C is bisected by the straight line CD meeting AB in D. Show that the rectangle BA, BD is, equal to the square on AD. 3. Similar triangles are to one another in the dupli cate ratio of their homologous sides. 4. It is required to draw a straight line perpendicular to a given plane from a given point without it. Show that this perpendicular is the shortest line that can be drawn from the point to the plane. 5. Solve the simultaneous equations 2002 + y2 = 171 3.x2 + xy + y2 = 27) 7. A line is divided into n parts, so that each part is times the part immediately to its left. The length of the extreme left and right parts being a and b respectively, find the length of the whole line in terms of a, b, and n. 8. Prove that the number of permutations of n things, r at a time, is n(n − 1)....(n − 1 + 1). In how many ways can three sets, consisting of 4, 5, and 6 things, be chosen from 16? 9. Define the cotangent of an angle, and trace the changes in its value as the angle increases from 0° to 360°. 10. Prove that cos (A + B) = cos A cos B A + B being less than 90°. sin A sin B, 11. Prove that cos (A + B + C) = cos A cos B cos C 4 cos (A + B C) cos (A B + C) |