determine what portions of their circumferences have been in contact. (17) One point describes the diameter AB of a circle with constant velocity, and another the semi-circumference AB from rest with constant tangential acceleration; they start together from A and arrive together at B; shew that the velocities. at B are as π : 1. (18) In the example of § 33 find in the case of e<1 the length of time occupied in the pursuit. (19) In the example of § 34 find the greatest distance the boat is carried down the stream, and shew that when it is in that position its velocity is √√/(u2 — v2). When u = parabola. v, shew directly that the curve described is a (20) Shew that if p be the radius of curvature of the curve of pursuit, we have in the figure of § 33, (21) In the case of a boat propelled with velocity u relatively to the water in a stream running with velocity v, shew that the boat passes from one given point to another in the least possible time when its actual path is a straight line. (22) The velocity of a stream varies as the distance from the nearest bank; shew that a man attempting to swim directly across will describe two semiparabolas. (Shew that the sub-normal is constant.) Find by how much the mean velocity is increased. (23) A point moves with constant velocity in a circle; find an expression for its angular velocity about any point in the plane of the circle. (24) If the velocity of a point moving in a plane curve vary as the radius of curvature, shew that the direction of motion revolves with constant angular velocity. (25) Two bevelled wheels roll together; having given the inclinations of the axes of the cones, find their vertical angles that the wheels may revolve with angular velocities in a given ratio. (26) Supposing the Earth and Venus to describe in the same plane circles about the Sun as centre; investigate an expression for the angular velocity of the Earth about Venus in any position, the actual velocities being inversely as the square roots of their distances from the Sun. (27) A particle moving uniformly round the circular base of an oblique cone is projected by generating lines on a subcontrary section; find its angular velocity about the centre of the latter. η (28) If E, n denote the co-ordinates of a moving point referred to two axes, one of which is fixed and the other rotates with constant angular velocity w, prove that its component accelerations parallel to these axes are (29) Two lines are moving in their own plane about their point of intersection with constant angular velocities w, w; if the co-ordinates of a moving point referred to them be x, y at a time t, prove that its accelerations parallel to the (30) Employ the formulæ of § (30) to trace approximately the form of the path of C about A, when m is nearly, but not exactly, equal to + n or to- n. (31) If an odd number n of rods OA,, 44, 4,4„...whose 1' 2 are hinged together at A1, A„...and n-1' revolve with constant angular accelerations a, 2x, 3z,...na, about their extremities 0,4,4,,...A,,, shew that the direction of motion of the point A at any time is perpendicular to the direction of the middle rod; the motion commencing from rest with the rods in a straight line. n (32) A man is in a boat, on a river, at a distance a from the shore, and b from a fall of water ahead. If the velocity of the stream be V, prove that he cannot escape the fall unless he can row with a velocity V; and that in case α √a2+b2 he can just row at this pace, the direction in which he must row is at right angles to the line joining his position with the point of the bank opposite the fall. Find also the direction in which he will have the least distance to row to reach the bank, supposing his velocity greater than this minimum. ✓ (33) If a point is moving in a hypocycloid with velocity u; and v, V represent the velocities of the centre of curvature and the centre of the generating circle corresponding to the position of the point, prove that c being the distance between the centres of the generating circles, and b the radius of the moving circle. (34) N particles are arranged equably along the circumference of a circle of radius a; each continually moves towards the next in order with a constant velocity v; shew that they will all arrive together at the centre of the circle in the time (35) A point P moves with constant velocity in a circle;✔ Qis a point in the same radius at double the distance from the centre, PR is a tangent at P equal to the arc described by P from the beginning of the motion: shew that the acceleration of the point R is represented in direction and magnitude by RQ. (36) If a point move in an orbit so that the area described in any time by the radius of curvature is proportional to that time, prove that the direction of the acceleration of the point is perpendicular to the line joining the point to the corresponding centre of curvature of the evolute, and its magnitude (F) is given by the equation where u is the index of curvature at the point, and c is twice the area described in a unit of time. (37) A body P is describing an ellipse in any manner : Qis a fixed point on the major-axis and PG the normal at P. Shew that at the moment when G coincides with Q, the angular velocity of P about Q is to its angular velocity about Gas CD to CB2. (38) A plane is moving about an axis perpendicular to it, and a point is moving in a given curve traced on the plane; in any position w is the angular velocity of the plane, v the velocity of the particle relative to the plane, r its distance from the axis, p the perpendicular on the tangent, s the arc described along the plane; prove that the acceleration along the tangent to the curve is (39) A particle moves on a surface: v, v′ are the components of its velocity along the lines of curvature, p, p' the principal radii of curvature; prove that the acceleration along v2 v2 the normal to the surface + 7. P ρ (40) The intrinsic equation of a curve being s=ƒ(4), the curve is described by a point with accelerations XY parallel to the tangent and normal at the point for which &=0; prove that cos 4 —; αφ аф (41) Obtain expressions for the accelerations of a moving point whose co-ordinates are r, 0, 4, (1) in the direction of r, (2) in the direction perpendicular to the radius vector and in the plane of 0, (3) in the direction perpendicular to the plane of 0. A point describes a rhumb line on a sphere in such a way that its longitude increases uniformly; prove that the resultant acceleration varies as the cosine of the latitude, and that its direction makes with the normal an angle equal to the latitude. (42) A rigid plane sheet is deprived by guide-pieces of all freedom of motion save parallel to a fixed line in its plane. If it be set in motion by the end of a crank, describing a given path in a given manner and working in a slot of given form cut in the sheet, form the equation of rectilinear motion of the sheet. (43) Investigate completely the cases of Example (42) when (a) the slot is straight, (b) the slot is a circular arc, the motion of the crank being circular and uniform. |