A block slides on a horizontal table, the coefficient of friction between them being 0\(\cdot\)2. The block is connected to a light thin string which passes over a small frictionless pulley fixed above the platform: the other end of the string is connected to a mass, hanging freely, immersed in a vessel of thick oil which renders all motion very slow. Initially the string is taut and makes an angle of 30\(^\circ\) with the horizontal: when motion ceases automatically, the angle of the string is 60\(^\circ\) with the horizontal. Plot a graph connecting the force of friction on the block with the displacement of the block; and hence shew that about 23\% of the total heat generated is generated in the vessel of oil.
The effective tractive force acting on a car of mass 1 ton which starts from rest is initially 350 lb. wt. It decreases uniformly (1) at the rate of 7 lb. wt. for every 10 feet travelled, (2) at the rate of 14 lb. wt. every second. Show that the velocity after the car has run 200 feet in the first case will be equal to its velocity after 10 secs. in the second case. [\(g=32\).]
Prove that the condition that the origin should lie on an asymptote of the conic \[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \] is \[ af^2 + bg^2 = 2fgh. \]
The tangent at a point \(P\) of the circle on the minor axis of an ellipse as diameter cuts the ellipse in \(Q, R\), and the tangents at \(Q\) and \(R\) meet in \(T\). Prove that \(PT\) is parallel to the major axis.
Atwood's machine consists of two masses attached to the ends of a light string which passes over a pulley. Two such machines are hung over a fixed pulley by means of a light string attached to the pulleys. Determine the motion, and prove that the pulleys move as a simple Atwood's machine, the mass of each pulley being increased by twice the harmonic mean of the masses of the particles connected with it. A simple Atwood's machine is carried on a truck, which slides down an inclined plane of angle \(\alpha\). Find the tension in the string and the other circumstances of the motion (i) when the plane is smooth, (ii) when there is a constant frictional retarding force.
The generating plant of an electric power station has an efficiency of 16\% at full load, viz. 600 kilowatts. The coal consumption at ``no load'' is one quarter of that at full load, and the consumption varies with load according to a straight line law; the output of the station for 24 hours is approximately divided as follows:
A pile of mass \(M\) is driven into the ground a distance \(a\) by means of a mass \(m\) falling on it from a height \(h\). Show that the average resistance of the ground is \[ (M+m)g + \frac{m^2gh}{(M+m)a}. \] If the pressure exerted between the mass and the pile during the impact increases uniformly from zero to a maximum and then decreases uniformly to a pressure which may be assumed to be zero, the whole duration of this process being a time \(t\) secs., so small that the pile has not sunk appreciably into the ground before it is completed, show that the maximum value of the pressure is \[ \frac{2Mm\sqrt{(2gh)}}{(M+m)t}. \]
Prove that there are in general four points on a conic such that the tangent at each point and the line joining it to a given point \(C\) (not on the conic) divide any given straight line \(AB\) harmonically. Prove also that these four points lie on a conic which divides harmonically both \(AB\) and the chord of the conic on which \(A, B\) lie, and passes through \(C\) and the pole of \(AB\).
Find the positions and magnitudes of the axes of the conic \[ 6x^2 + 4xy + 9y^2 - 20x - 10y + 15 = 0. \]
Prove the following properties of the centre of mass: