2556. A straight tube inclined to the vertical at an angle a revolves with uniform angular velocity w about a vertical axis whose shortest distance from the tube is a and contains a smooth heavy particle which is initially placed at its shortest distance from the axis: prove that the spaces which the particle describes along the tube in a time t is given by the equation 2557. A heavy particle is attached to two points in the same horizontal plane at a distance a by two extensible strings each of natural length a, and is set free when each string is at its natural length: prove that the radius of curvature of the initial path of the particle is 2√3a ÷ (m ~ n), the moduli of the strings being respectively m and n times the weight of the particle. 2558. Three equal particles P, Q, Q', for any two of which e= 1, move in a smooth fine circular tube of which AB is a vertical diameter; P starts from A, and Q, Q' at the same instant in opposite senses from B, the velocities being such that at the first. impact all three have equal velocities prove that throughout the whole motion the straight line joining any two particles is either horizontal or passes through one of two fixed points (images of each other with respect to the circle); and that the intervals of time between successive impacts are all equal. 2559. A point P describes the curve y = a log sec with a velocity a which varies as the cube of the radius of curvature and has attached to it a particle by means of a string of length a; when P is at the origin, is at the corresponding centre of curvature and its velocity is equal and opposite to that of P: prove that throughout the motion the velocity of will be equal in magnitude to that of P, and that Q is always the pole of the equiangular spiral of closest contact with the given curve at P. IV. Motion of Strings on Curves or Surfaces. 2560. A uniform heavy chain is placed on the arc of a smooth vertical circle, its length being equal to that of a quadrant and one extremity being at the highest point of the circle: prove that in the beginning of the motion the resultant vertical pressure on the circle bears to the resultant horizontal pressure the ratio 2 - 4 : 4. 2561. A string of variable density is laid on a smooth horizontal table in the form of a curve such that the curvature is everywhere proportional to the density and tangential impulses are applied at the ends prove that the equation for determining the impulsive tension T at any point is T = Ae + Be ̄, where p is the angle which the tangent makes with a fixed direction; and that, if the curve be an equiangular spiral, the initial direction of motion of any point will be at right angles to the radius vector. 29... 29 2562. A number of material particles P1, P, of masses m,, m connected by inextensible strings are placed on a horizontal plane so that the strings are sides of an unclosed polygon each of whose angles is – a, and an impulse is applied to P, in the direction PP, prove π that 1 where T, is the impulsive tension of the 7th string; and deduce the equation 1 dp dT p2 2 for the impulsive tension in the case of a fine chain. From either equation deduce the result of the last question. 2563. A fine chain of variable density is placed on a smooth horizontal table in the form of a curve in which it would hang under the action of gravity and two impulsive tensions applied to its ends, which are to each other in the same ratio as the tensions at the same points in the hanging chain: prove that the whole will move without change of form parallel to the straight line which was vertical in the hanging chain. 2564. A heavy uniform string PQ, of which P is the lower extremity, is in motion on a smooth circular arc in a vertical plane, O being the centre and OA the horizontal radius: prove that the tension at any point R of the string is where 0, 2a, 2y are the angles AOP, POQ, POR respectively, and W the weight of the string. 2565. A portion of a heavy uniform string is placed on the arc of a four-cusped hypocycloid, occupying the space between two adjacent cusps, and runs off the curve at the lower cusp where the tangent is vertical: prove that the velocity which the string will have when just leaving the arc will be that due to a space of nine-tenths the length of the string. 2566. A uniform string is placed on the arc of a smooth curve in a vertical plane and moves under the action of gravity: prove the equation of motion 7 being the length of the string, s the arc described by any point of it at a time t, and y,, y, the depths of its ends below a fixed horizontal straight line. 2567. A uniform heavy string APB is in motion on a smooth curve in a vertical plane, and on the horizontal ordinate from a fixed vertical line to A, P, B are taken lengths equal to the arcs measured from a fixed point of the curve to A, P, B respectively: prove that the ends of these lengths are the corners of a triangle whose area is always proportional to the tension at P. 2568. A uniform heavy string is placed on the arc of a smooth cycloid whose axis is vertical and vertex upwards: determine the motion, and prove that, so long as the whole of the string is in contact with the cycloid, the tension at any given point of the string is constant throughout the motion and greatest at the middle point (measured on the arc). 2569. A uniform heavy chain is in motion on the arc of a smooth curve in a vertical plane and the tangent at the point of greatest tension makes an angle with the vertical: prove that the difference between the depths of the extremities is l cos p. 2570. A uniform inextensible string is at rest in a smooth groove, which it just fits, and a tangential impulse P is applied at one end: prove that the normal impulse per unit of length at a distance s (along the arc) from the other end is Ps÷ap, where a is the whole length of the string and p the radius of curvature at the point considered. 2571. A straight tube of uniform bore is revolving uniformly in a horizontal plane about a vertical axis at a distance e from the tube, and within the tube is a smooth uniform chain of length 2a which is initially at rest with its middle point at the distance c from the axis of revolution: prove that the chain in a time t will describe a space c(ewt-e-wt) along the tube, and that the tension of the chain at a point distant x from its middle point is where m is the mass of the chain and w the angular velocity. 2572. A circular tube of radius a revolving with uniform angular velocity about a vertical diameter contains a heavy uniform rigid wire which just fits the tube and subtends an angle 2a at the centre: prove that the wire will be in relative equilibrium if the radius to its middle point make with the vertical an angle whose cosine is g÷ aw2 cos a, and that the stress along the wire is a minimum at the lowest point of the tube (provided the wire pass through that point) and a maximum at the point whose projection on the axis bisects the distance between the projections of the ends of the wire. Discuss which position of equilibrium is stable, proving the equation of motion d20 where is the angle which the radius to the middle point of the wire makes with the vertical. [The highest position of equilibrium is always unstable; the oblique position is stable if it is possible, the time of a small oscillation being and the lowest position is stable when aw2 cos a <g, the time of a small oscillation being 2573. A pulley is fixed above a horizontal plane; over the pulley passes a fine inextensible string, which has two equal uniform chains fixed to its ends; in the position of equilibrium a length a of each chain is vertical, and the rest is coiled up on the table. One chain is now drawn up through a space na: form the equation of motion, and prove that the system will next come to instantaneous rest when the upper end of the other chain is at a depth ma below its mean position, where (1 − m) €" = (1 + n) e ̄”. Also, when n = 1, prove that m -- 5623 nearly. V. Resisting Media. Hodographs. 2574. A heavy particle is projected vertically upwards, the resistance of the air being mass (velocity)÷c; the particle in its ascent and descent has equal velocities at two points whose respective heights above the point of projection are x, y: prove that 2575. A heavy particle moves in a medium in which the resistance varies as the square of the velocity, v, v', u are its velocities at the two points where its direction of motion makes angles -4, with the horizon and at the highest point, and p, p', r are the radii of curvature at the same two points and the highest point respectively: prove that 2576. A heavy particle moves in a medium whose resistance varies as the 27th power of the velocity; e, v', u are the velocities of the particle when its direction of motion makes angles - 4, with the horizon and at the highest point, and p, p', r are the radii of curvature at the same two points and the highest point respectively: prove that 28 с 2577. A small smooth bead slides on a fine wire whose plane is vertical and the height of any point of which is a sin 8 being the arc measured from the lowest point, in a medium whose resistance is mass × (velocity) c, and starts from the point where 8s πс: prove the velocity acquired in falling to the lowest point is. = 2578. A heavy particle slides on a smooth curve whose plane is vertical in a medium whose resistance varies as the square of the velocity, and in any time describes a space which is to the space described in the same time by a particle falling freely in vacuo as 12n: prove that the curve must be a cycloid whose vertex is its highest point, and that the starting point of the particle must divide the arc between two cusps in the ratio 2n-1 : 2n + 1. 2579. A heavy particle falls down the arc of a smooth cycloid whose axis is vertical and vertex upwards in a medium whose resistance is mass (velocity) 2c, its distance along the arc from the vertex being initially c: prove that the time to the cusp will be √sa (ta -1) where 2a is the length of the axis. 2580. A particle is projected from a fixed point A in a medium. whose resistance is measured by 3 x velocity and attracted by a fixed point S by a force whose acceleration is 2wx distance: prove that the particle will describe a parabola tending in the limit to come to rest at S. [Taking SA = a, and u, v to be the component velocities at A along and perpendicular to SA, the equation of the path is {(u+aw) y - vx} = av {vx - (u + 2aw) y}, and the length of the latus rectum is = 2581. A heavy particle moves in a circular tube whose plane is vertical in a medium whose resistance is mass x (velocity) ÷ 2c, starting from a point in the upper semicircle where the normal makes the angle tan' with the vertical: prove that the kinetic energy с a at any time while the particle moves through the semicircle which begins at this point is proportional to the distance of the particle from the bounding diameter. 2582. A point describes a straight line under acceleration tending to a fixed point and varying as the distance: prove that the corresponding point of the hodograph will move under the same law of acceleration. 2583. The curves" = a” sin m0, r" = a" sin n✪ will be each similar to the hodograph of the other when described about a centre of force in the pole, provided that mn+m + n = 0. Prove this property geo metrically for both curves when m=1, 2n=-1. 2584. A point describes a certain curve in such a manner that its hodograph is described as if under a central force in its pole, and T, N are the tangential and normal accelerations of the point: prove that |