2. Shew that for the equilibrium of a rigid body acted on by any forces in a plane it is necessary and sufficient that the sums of the resolved parts of the forces in two directions and the moment of the forces about one point vanish. A heavy disc is free to move about a point A in its plane. A string is attached to the centre of gravity G, and pulled horizontally with a force T. If W be the weight of the disc, shew that the tangent of the angle between A G and the vertical is Τ W 3. Find the centre of gravity of a uniform triangular lamina. An equilateral triangle and a square have a side in common. Find the distance of the centre of gravity of the two from the centre of the square. 4. Give the equations connecting (1) the dynamical measure of the force on a particle with its mass and acceleration; (2) the force in pounds and poundals; (3) the force in foot pound second units, and in centimetre gramme second units. [Take a kilogramme as 2.2 lbs., and a metre as 39 37 inches.] 5. Prove the equation s = vt + ft2 for motion in a straight line under constant acceleration f, v being the velocity at time zero, and s the space passed over. Shew that a stone thrown a height of 200 feet will be within 72 feet of the ground on the way down in 6 4 seconds nearly from the time of its leaving the hand. 6. Prove that the acceleration of a particle moving uniformly in a circle of radius r with velocity v is towards the centre, and equals v2 Hence shew that, apart from the resistance of the air, a particle would revolve around the earth close to its surface in about 1 h. 25 min. [Take 4,000 miles as the radius of the earth and g = 32.] 7. Demonstrate the equation pgpz for the pressure at the depth z in a liquid of density p. Shew that the pressure on a square foot at a depth of 1 mile below the surface of water is about 147 tons. 8. State and prove Archimedes' theorem relative to the apparent loss of weight of a body immersed in a liquid. A solid mass of 12 lbs. weighs 3 lbs. in one liquid and 4 lbs. in another. Compare the densities of the liquids. 9. Explain carefully the mode of action of the common pump. MIXED MATHEMATICS.-PART II. The Board of Examiners. TO BE USED ALSO AS FIRST HONOUR PAPER. 1. Prove that two couples in the same plane are equivalent if their moments are the same. 2. Investigate the reduction of a system of coplanar forces to two forces ΣΧ, ΣΥ parallel to the rectangular axes x, y, and a couple Σ(yX— x Y). 3. State Coulomb's Laws of Friction. A uniform heavy isosceles triangular lamina ABC is supported on small studs at its corners, and rests on an inclined plane, with its base BC horizontal, the coefficient of friction of the stud A being μ, and of B and C με Shew that slipping takes place when the elevation of the plane is 3 4. Find the centre of gravity of a tetrahedron. Shew that the centre of gravity of a slice cut off a spherical shell of any thickness by two parallel planes is midway between the planes, provided the slice has a hole through it. 5. Shew that the components of acceleration of a point (x, y) referred to rectangular coordinates are d2x d'y 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 V 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 from rest at distance a to the centre is π μ a3 8μ where is the astronomical mass of the attracting centre. |