Establish that the radius of curvature of a plane curve whose pedal equation is \(r=r(p)\) is \(r dr/dp\). Show that the square of the length of the tangent from the pole to the circle of curvature at a general point is given by \(\frac{d}{dq}(r^2q)\), where \(q=1/p\). Hence show that if a curve is such that all its circles of curvature pass through a certain fixed point, then it must itself be a circle.
A plane framework is constructed of seven equal light inextensible rods, \(AB, AC, BC, BD, CD, CE, DE\), freely hinged together so as to form the equilateral triangles \(ABC, BCD, CDE\). It is freely hinged at \(A\) to a vertical wall so that \(BD\) and \(ACE\) are horizontal, with \(BD\) uppermost, and a chain \(FB\), of length equal to \(AB\), connects \(B\) to a point \(F\) of the wall vertically above \(A\). Loads of 60 lb. hang from \(C\) and \(E\). Find, by calculation or by drawing, the forces in the rods, indicating which of the forces are tensions.
Two equal uniform cubes, each of weight \(W\), stand on a horizontal table with a small gap between them, and with the line joining their centres perpendicular to adjacent vertical faces. A light wedge of semi-angle \(\beta (<\frac{1}{4}\pi)\) lies symmetrically, vertex downwards, between the cubes and is acted on by a slowly increasing vertical force \(P\) until equilibrium is broken. The coefficient of friction between either cube and the table is \(\mu (<\cot\beta)\), and there is no friction between the wedge and either cube. Find the values of \(P\) for which equilibrium would be broken on the assumption (a) that the cubes slide, (b) that they topple outwards. Hence show that the equilibrium is broken by the cubes sliding if \(\mu < 1/(2-\tan\beta)\).
Show that the centre of mass of a sector, of angle \(2\alpha\), cut from a uniform thin circular disc of radius \(a\), is distant \((2a\sin\alpha)/3\alpha\) from the centre of the circle. A radial cut \(OA\) is made in a uniform thin circular disc, of centre \(O\) and radius \(a\). A quadrantal portion \(AOB\) is folded over so that \(AOB\) lies in contact with the other portion of the disc. Neglecting the thickness of the disc, find the distance from \(O\) of the centre of mass of the folded disc.
Two equal straight light rods \(AB, BC\), each of length \(l\), are freely hinged together at \(B\), where they carry a mass \(M\); masses \(m\) are borne at \(A\) and at \(C\). The rods rest symmetrically across two smooth parallel horizontal rails, of small cross-section, fixed at a distance \(2c\) apart and at the same level. Show that a position of equilibrium exists in which the plane of the rods is perpendicular to the rails and \(B\) is at a higher level than \(A\) and \(C\), provided that \(M\) is less than \(2m(l-c)/c\). Show that if this position of equilibrium exists it is stable for displacements in which the plane of the rods remains perpendicular to the rails and \(B\) is displaced vertically. Investigate the corresponding problem with \(B\) below \(AC\).
An air race is flown over a course in the shape of an equilateral triangle \(ABC\), in which \(B\) is due east of \(A\) and \(C\) is north of \(AB\). An aeroplane flies over the course at a constant level and at constant speed relative to the wind, which may be assumed not to vary. If the measured speeds along \(AB, BC, CA\) are \(v_1, v_2, v_3\) respectively, calculate the east component (\(U\)) of the velocity of the wind.
A fire-pump is raising water from a reservoir 50 ft. below the nozzle and delivering in a jet 4 in. in diameter with nozzle-velocity 60 ft./sec. Taking \(g\) to be 32 ft./sec.\(^2\), \(\pi\) to be 22/7 and the density of water to be \(62\frac{1}{2}\) lb/ft.\(^3\), find approximately the horse-power at which the pump is working. Frictional resistances are supposed neglected. If the jet impinges horizontally on a vertical wall and does not rebound, find in lb. wt. the force exerted on the wall.
A man, whose weight is 150 lb., is standing on a rung of a ladder near the top of a mast of a ship which is rolling with a period of 10 sec. Regarding the man as moving horizontally with simple harmonic motion of amplitude 8 ft. on either side of the vertical, find the total horizontal force that the man must be able to exert in order not to be thrown off. Find his horizontal speed and his displacement from his mean position at an instant when he is exerting a horizontal force of 10 lb. wt.
Apply the principles of the conservation of energy and angular momentum to solve the following problem: A light inextensible string \(AB\) passes through a small hole \(O\) in a smooth horizontal table and has particles of equal masses fastened at its two ends. Initially the mass at \(A\) on the table is held at rest and the mass at \(B\) hangs at rest. If the length of the horizontal portion of the string \(OA\) is initially \(r_0\) and the mass at \(A\) is projected horizontally at right angles to \(OA\) with velocity \((2gr_0)^{\frac{1}{2}}\), find the length of string on the table when \(B\) is next instantaneously at rest.
Three equal smooth billiard balls \(A, B, C\), are at rest on a smooth horizontal table with their centres in a straight line. The ball \(A\) is projected directly towards \(B\) with velocity \(U\), and after the impact \(B\) goes on to strike \(C\). If the coefficient of restitution for all impacts is \(e\), find the velocity with which \(C\) moves off. Investigate whether \(A\) will strike \(B\) a second time and whether \(B\) will subsequently strike \(C\) again.