A dynamo, of E.M.F. 105 volts and internal resistance 0.025 ohm, is in parallel with a storage battery of E.M.F. 100 volts and internal resistance 0.06 ohm. They are feeding an external circuit of resistance 1.75 ohm: find whether the battery is charging or discharging, and calculate the current through the dynamo and the P.D. at the terminals of the external circuit.
Using areal (or trilinear) coordinates, find the coordinates of the centre of a conic circumscribing the triangle of reference. \par Two conics circumscribe a triangle and touch one another at one of the angular points. Prove that their two centres, their point of contact, and the middle points of the sides of the triangle lie upon a conic.
A spring of negligible inertia carries a pan weighing 1 ounce, and is such that a \(\frac{1}{2}\) lb. weight will lower the pan by 1 inch. It is compressed 2 inches and placed on a table with its axis vertical: a 2 ounce weight is put on it and the spring released. Find how high the weight rises before it leaves the spring and its velocity at that instant.
Prove that the middle points of a system of parallel chords of the curve \[ ax^2+2hxy+by^2=1 \] lie on a straight line through the origin. \par Show that the chord of this curve which has \((X, Y)\) for its middle point is \[ axX + h(xY+yX)+byY = aX^2+2hXY+bY^2. \]
Define a unit magnetic pole. How is a ``line of magnetic force'' defined by means of the unit pole? \par Two bar magnets, each of length 50 cms. and pole strength 50 units, are laid centrally across one another at right angles. Find the couple in dyne-centimetres on each magnet due to the other.
A segment of a circle is to have a given area, and the length of the chord of the segment together with \(n\) times the length of the arc is to be a minimum. Prove that if \(n>1\) the segment must be greater than a semicircle, and that the angle in the segment must have its secant equal to \(n\). \par What is the solution if \(n<1\)?
A column of water 30 feet long is moving behind a plug piston in a pipe of uniform diameter, with a velocity of 15 feet per second. Prove that the time average of the pressure of the water on the piston, caused by its stoppage in one-tenth of a second, is 610 lbs. per square inch.
Express the area \(S\) of a triangle in terms of the lengths of the sides. \par Prove that \[ \frac{\partial S}{\partial a} = R \cos A, \] where \(R\) is the radius of the circumcircle.
Prove that, if \(y\) is equal to \(e^x\), or if \(y\) is equal to the sum of the first \(n+1\) terms of the expansion of \(e^x\) in ascending powers of \(x\), \[ x \frac{d^2y}{dx^2} - (n+x)\frac{dy}{dx} + ny = 0. \]
Water issues vertically from the nozzle of a fire hose, the sectional area of which is one square inch, with a velocity of 130 feet per second. Find the discharge in cubic feet per second, and the horse-power of the pump engine, assuming the efficiency to be \(70\%\), and that the nozzle is 50 feet above the pump.