Problems

A dynamo giving a terminal P.D. of 140 volts is used to charge a battery of 55 cells in series, each giving a back \textsc{e.m.f.} of 2\(\cdot\)2 volts and having a resistance of 0\(\cdot\)002 ohm. If the charging current required be 30 amperes, find the extra resistance which must be put in the circuit, and make out a balance sheet, showing on the one side the total output of the dynamo, and on the other side the separate items in the power account.

A submarine which travels at 10 knots sights a steamer 12 nautical miles away in a direction 40\(^\circ\) West of South. The steamer is travelling at 15 knots due N. Show graphically that there are two directions in which the submarine can proceed so as to intercept the steamer, and calculate the least time in which it can do so.

\(ABC\) is an acute-angled triangle, \(D, E, F\) are the middle points of the sides \(BC, CA, AB\) respectively, and \(O\) is the circumcentre. On \(OE, OF\) produced points \(Q, R\) respectively are taken so that the angles \(CQA, ARB\) are supplementary. Prove that \(DQ, DR\) are perpendicular.

Prove that if an observer at height \(h_1\) above the earth's surface can see a fixed object at height \(h_2\), the observer must be somewhere within a region of area \[ 2\pi R(h_1 + h_2 + 2\sqrt{h_1h_2}) \] approximately, where \(R\) is the radius of the earth.

Establish the following theorems, deducing (3) as a consequence of (1).

1. [(1)] If forces acting at a point \(O\) are represented by \(p \cdot OP, q \cdot OQ, r \cdot OR \dots\) their resultant is represented by \((p+q+r+\dots)OG\), where \(G\) is the centroid of particles of masses \(p, q, r \dots\) placed at \(P, Q, R \dots\), respectively.
2. [(2)] If a system of \(n\) forces is represented in all respects by lines \(A_1B_1, A_2B_2, \dots\) their resultant is represented in magnitude and direction by \(n \cdot G_1G_2\), where \(G_1\) is the centroid of equal particles placed at \(A_1, A_2, \dots\), and \(G_2\) is the centroid of equal particles placed at \(B_1, B_2, \dots\).
3. [(3)] The line of action of the resultant of two parallel forces is parallel to them, and divides any line joining two points, one on each of their lines of action, in the inverse ratio of their magnitudes, the point of division being internal if the forces are like.

Describe briefly with sketches three common types of voltmeter. State the peculiar advantages and disadvantages of each type. Place the three types in your estimated order of sensitiveness as measured by the scale deflection obtained with a given expenditure of power, and explain the fundamental reasons for expecting this order.

The case of a rocket weighs 1 lb. and the charge weighs 4 lb. The charge burns at a uniform rate and is completely burnt in 2 secs., and during that time exerts a constant propulsive force equal to 20 lb. weight. If the rocket is fired vertically, find by plotting the acceleration time curve, or otherwise, the vertical velocity acquired during the burning of the charge. [Log\(_{10}e = \cdot 43\).]

The normals to an ellipse at the ends of a variable chord through a fixed point meet in \(P\); prove that the straight line through the feet of the remaining normals from \(P\) envelopes a parabola.

Prove that the least positive root of the equation \[ x = 2\pi \sec x \] is \(2\pi\), and that there is one other root between \(2\pi\) and \(\frac{5}{2}\pi\). Determine approximately the numerical value of this other root.

Discuss by means of two or three illustrations the meaning of potential energy. Shew that the potential energy is stationary in an equilibrium position of a statical system, and distinguish between stable and unstable positions, justifying your statements. A uniform rectangular beam of depth \(2h\) rests in a horizontal position on the top of a rough circular cylinder of radius \(a\), whose axis is horizontal and perpendicular to the longer edges of the beam. If \(a>h\), shew that the beam may be rotated by rolling on the cylinder into an unstable inclined position of equilibrium. Shew also that the horizontal position of the beam is unstable if \(a=h\), and that in this case the inclined position of equilibrium does not exist.