1 FACULTY OF PRACTICAL SCIENCE. MATHEMATCS III. 1. Differentiate sin 0 (giving work in full), tan1(3x-2), (x sin-1x), e2* log√1 −x, (1 − 2x)/cos*x 2. Find the angle between xy=6 and y2=3x at their point of intersection. 3. Find the dimensions of an open tank with a square base and vertical sides, whose capacity is 900 cub. ft., such that the cost of lining it with lead shall be least. dy (x-x1) as the equation of the 4. Derive v-Y1 = tangent to the curve y=f(x) at the point (x1,y1). From this deduce the equation of the normal at this point. 7. Employ the calculus to find the volume of the solid formed by a circle of 2" radius rotating about a line tangent to it. 8. Solve √a? —y1dy=ydx, and determine the constant of integrat'on so that y=a when x=0. 9. Find the moment of inertia about the x-axis of the parabolic sector bounded by y2=4x and the line x=6. FACULTY OF PRACTICAL SCIENCE. PHYSICS I (A). All data is to be taken as correct to one per cent. and the problems are to be worked out to numerical results accordingly. 1. A waggon moves along a straight road at 5 miles an hour. The front and rear wheels are respectively 2.5 and 3 feet in diameter. (a) Find the angular velocity of the front wheels relatively to that of the rear wheels. (b) If at a given instant a spoke of the front wheel be parallel to a certain spoke in the rear wheel, find how long it will be before these two spokes are perpendicular to one another. 2. An old watch (mass 80 grams) is hung up by passing through its ring the cords that support a small picture. The picture is then hung from a nail and the watch slips down until the portions of the cords running from the ring to the corners of the picture make an angle of 60° with one another, while the parts of the cords above the watch are both vertical. Neglecting friction, find the mass of the picture. 3. A man (mass 180 lbs.) sits centrally on a light office stool opposite the desk in an express car. His centre of gravity is 4 feet above the floor and the legs of the stool rest on the corners of a square of 18 inch side (two sides being parallel to the track beneath the car). The brakes are put on and the train is stopped from 45 miles per hour in 800 feet. (a) Find the vertical reactions between each leg of the stool and the floor; (b) Find also the total horizontal force between the legs and the floor. 4. In the California Street cable road at San Francisco, the horse-power required to haul the cable alone at 6 miles per hour is 84. (a) Find, in pounds weight, the resistance overcome. (b) If each cable car offer a frictional resistance of 150 lbs. wt., and if the rate of working on the cable be unaltered, how many cars could grip the cable at the same time before the speed would be reduced to 4 miles per hour? 5. A uniform ladder rests on a wooden floor (coefficient of friction, 0.4) and against a brick wall (coefficient of friction, 0.5). Express, as the inverse tangent, the least angle to the horizontal at which the ladder will remain without slipping. 6. A man throws a block of wood (mass 1 kilogram) 10 metres straight up into the air. It reaches its highest position and after it has fallen 4.9 metres a second man shoots into it a 10 gram bullet with a velocity of 30,000 centimetres per second at an angle of 45° to the horizontal. The bullet beds itself in the block. (a) Find the horizontal and vertical components of the momentum immediately after the impact. stant. (b) Deduce the components of the velocity at this in (c) Express, as a complex fraction, the horizontal distance through which the impact displaces the block in its fall to the ground. PHYSICS I (B). Experimental Subjects. Answer any nine of the twelve questions. 1. Define :-electromagnetic unit current, British thermal unit, potential difference, magnetic permeability, focal length of a lens, boiling point. What is a chromatic abberation and how is it remedied? 2. A car is lighted by 5 lamps of 220 ohms resistance each, joined in series. If the difference in potential across the ends of the lamp circuit be 550 volts, what current flows through the lamps? What power is expended in this circuit, and what does it cost to light the car at 9c. per kilowatt hour? 3. The diameter of a circular coil of 10 turns of wire is 30 cms., and its resistance o.1 ohm. Find the quantity of electricity in coulombs which will flow in the coil when revolved from a position at right angles to a magnetic field to a position parallel to the field. The strength of the field is 20 c.g.s. units. 4. Describe the mode of operation of an induction coil with a sketch of its parts. What is a transformer and what does it transform? What is an alternating current and how is it obtained? 5. Explain in detail one method of measuring the wave length of light and derive the formula for that case. 6. The diameter of the sun is of its distance from the earth. How many inches in diameter will be the image of the sun with a concave spherical mirror whose radius is 96 feet? |