103. To find 7, the time of oscillation, when the amplitude of oscillation is 2a, 104. The above solution fails when n = 1, but the time of falling to the centre may be found as follows. The equation for this case, as given in § 94, is the negative sign being taken since a diminishes as t increases. Put T for the required time, then 105. A particle is constrained to move in a straight line, and is acted on by an attraction directed to a point not in ⚫ that line, and expressed by a function (r) of the distance; to determine the time of a small oscillation. Employing the same notation as in § 99, the acceleration along PO being $(7), its component along PN is ø (r) ==, therefore the equation of motion is and therefore by § 90, the time of a small oscillation is (1) A body is projected vertically upwards with a velocity which will carry it to a height 2g; shew that after three seconds it will be descending with a velocity g. (2) Find the position of a point on the circumference of a vertical circle, in order that the time of rectilinear descent from it to the centre may be the same as the time of descent to the lowest point. (3) The straight line down which a particle will slide in the shortest time from a given point to a given circle in the same vertical plane, is the line joining the point to the upper or lower extremity of the vertical diameter, according as the point is within or without the circle. (4) Find the locus of all points from which the time of rectilinear descent to each of two given points is the same. Shew also that in the particular case in which the given points are in the same vertical, the locus is formed by the revolution of a rectangular hyperbola. (5) Find the line of quickest descent from the focus to a parabola whose axis is vertical and vertex upwards, and shew that its length is equal to that of the latus rectum. (6) Find the straight line of quickest descent from the focus of a parabola to the curve when the axis is horizontal. (7) The locus of all points in the same vertical plane for which the least time of sliding down an inclined plane to a circle is constant is another circle. (8) Two bodies fall in the same time from two given points in space in the same vertical down two straight lines drawn to any point of a surface; shew that the surface is an equilateral hyperboloid of revolution, having the given points as vertices. (9) Find the form of a curve in a vertical plane, such that if heavy particles be simultaneously let fall from each point of it so as to slide freely along the normal at that point, they may all reach a given horizontal straight line at the same instant. (10) A semicycloid is placed with its axis vertical and vertex downwards, and from different points in it a number of particles are let fall at the same instant, each moving down the tangent at the point from which it sets out; prove that they will reach the involute (which passes through the vertex) all at the same instant. th (11) A particle moves in a straight line under the action of an attraction varying inversely as the power of the distance; shew that the velocity acquired by falling from an infinite distance to a distance a from the centre is equal to the velocity which would be acquired in moving from rest at a distance a to a distance a 4 (12) A particle moves in a straight line from a distance a towards a centre of attraction varying inversely as the cube of the distance; shew that the whole time of descent (13) A particle is placed at a given point between two centres of equal intensity attracting directly as the distance; to determine the motion and the time of an oscillation. Let 2a be the distance between the centres, x the distance of the particle at any time from the middle point between them, then the equation of motion is (14) If a particle begin to move directly towards a fixed centre which repels with an intensity =μ (distance), and with an initial velocity = μ (initial distance), prove that it will continually approach the fixed centre, but never attain to it. ✓ (15) A particle acted upon by two centres of attraction, each attracting with an intensity varying inversely as the square of the distance, is projected from a given point between them, to find the velocity of projection that the particle may just arrive at the neutral point of attraction and remain at rest there. If μ, μ' be the strength of the centres; a, a, the distances of the point of projection from them; and V the initial velocity; we have (μαμα) (16) Supposing the earth a homogeneous spheroid of equilibrium, the time of descent of a body let fall from any point P on the surface down a hole bored to the centre Č, varies as CP, and the velocity at the centre is constant. (17) A material particle placed at a centre of attraction varying as the distance, is urged from rest by a constant force which acts for one-sixth of the time of a complete oscillation about the centre, ceases for the same period, and then acts as before, shew that the particle will then be retained at rest, and that the distances moved through in the two periods are equal. (18) A body moves from rest at a distance a towards a centre of attraction varying inversely as the distance, shew that the time of describing the space between Ba and ẞ"a will be a maximum if ẞ= 1 1 T. D. n 2(n-1) 6 |