Define the hodograph. Shew that if \(P\) be a moving point and \(Q\) the corresponding point in the hodograph, the velocity of \(Q\) represents in magnitude and direction the acceleration of \(P\). Find the acceleration of a point moving uniformly in a circle. The weight of a truck is 10 tons and it is moving at a speed of 15 miles per hour round a curve of radius 4000 feet, which is banked up so that there shall be no lateral thrust on the rails at a speed of 20 miles per hour. Find the lateral thrust of the truck on the rails.
What is meant by the statement that ``the mechanical equivalent of a Thermal Unit in Pound-Centigrade units is 1400 ft. lbs.''? Describe any experiment for determining this ratio. A 30 H.P. petrol engine at full load consumes 21 lbs. of petrol per hour, the calorific value of the fuel being 9500 Pound-Centigrade Thermal Units per pound. Find the thermal efficiency of the engine. If the power is absorbed by a friction brake kept cool by a steady stream of water, which is supplied at 20\(^{\circ}\) C. and slowly boiled away, find how much water will be used per hour, if the latent heat of steam at atmospheric pressure be 536 Thermal Units.
Shew that (the coordinates being areal) the conditions that \(px+qy+rz=0\) should be an asymptote of \(2fyz + 2gzx+2hxy=0\) are \[ f:g:h :: p(q-r)^2 : q(r-p)^2 : r(p-q)^2. \]
Find an expression for the velocity at any point in the path of a particle moving with simple harmonic motion. After the particle is 3 inches from the middle point of the path, moving away from the middle point, 4 seconds elapse until it is again in that position, moving towards the middle point, whilst a further 10 seconds elapses until it again arrives at that position. Find the length of the path.
A point \(O\) moves on the line which bisects the angle \(C\) of a triangle \(ABC\), and \(AO, BO\) produced meet \(CB, CA\) in \(P\) and \(Q\) respectively. Prove that, as \(O\) approaches \(C\), the ratio \(CP:CQ\) tends to the limit unity, and as \(O\) approaches \(AB\) the ratio \(BP:AQ\) tends to the limit \(BC^2:AC^2\).
Distinguish between ``Potential difference'' and ``Electromotive force.'' A cell of E.M.F. 2 volts and internal resistance 1 ohm sends current through an external resistance of 10 ohms, whilst a voltmeter of 40 ohms resistance is put across the terminals of the cell. Find the reading of the voltmeter.
Shew that
\[
f(x) = \frac{1-x}{\sqrt{x}} + \log x
\]
has a differential coefficient which is negative for all values of \(x\) between 0 and 1. Hence shew that, if \(0
Two small rings of masses \(m, m'\) are moving on a smooth circular wire which is fixed with its plane vertical. They are connected by a straight massless inextensible string. Prove that, while the string remains tight, its tension is \(2mm'g \tan\alpha \cos\theta/(m+m')\), where \(2\alpha\) is the angle subtended by the string at the centre of the ring, and \(\theta\) is the inclination of the string to the horizon.
The base \(BC\) of a triangle \(ABC\) is fixed and the vertex \(A\) undergoes a small displacement in a direction inclined at an angle \(\theta\) to \(CA\) and at an angle \(\phi\) to \(BA\). Prove that the increments of the sides \(b, c\) and the angles \(B, C\) are connected by the relations \[ \delta b \sec\theta = \delta c \sec\phi = c \delta B \operatorname{cosec}\phi = b \delta C \operatorname{cosec}\theta. \]
Find the magnetic force at the centre of a circular coil containing 20 turns of radius 10 cm. when a current of 5 amperes is flowing. (A C.G.S. unit of current is 10 amperes.)