(9) Two particles, connected together by a rigid rod without weight, are projected along a smooth horizontal plane: to determine their motion.

Let the plane of co-ordinates coincide with the plane of the motion. Let m, n, be the resolved parts of the initial velocity of the centre of gravity of the two particles parallel to the axes of x, y, and let a, b, be its initial co-ordinates. Let w be the initial angular velocity of the rod, e its inclination to the axis of x at the end of the time t, and e at the beginning of the motion. Then the position of the centre of gravity is given at any time t by the equations

x = mt + a, y = nt + b ;

and the inclination of the rod to the axis of x, by the equation 0 = wt + €.

Clairaut; Mémoires de l'Académie des Sciences de Paris,

1736, p. 7. Euler; Act. Acad. Petrop. 1780, P. 1; Opuscula, De motu corporum flexibilium, Tom. III. p. 91.

(10) A spherical particle moves within a smooth tube, which revolves about one extremity with a uniform angular velocity in a vertical plane, the capacity of the tube being just sufficiently great for the reception of the particle to determine the motion of the particle.

Let Ox, (fig. 161), which is horizontal, be the initial position of the tube, and P the position of the particle in the tube after a time t. Let w denote the angular velocity of the tube about 0, the inclination of OP to Or, and let OP r. Then, supposing the initial velocity of the particle to be zero, and that_r=a initially, the value of r at any time t is given by the equation

=

and the polar equation to the path of the particle will result

from the substitution of for at in this equation. When t becomes very great, the polar equation becomes

which is the equation to an equiangular spiral.

The solution of this problem was attempted by M. Le Barbier, in the Annales de Gergonne, Tom. XIX. p. 285, who omitted to take into consideration the centrifugal force, an oversight which entirely vitiated his results. The correct solution was given in Tom. xx. by Ampère.

(11) A smooth wire in the form of a circle is made to revolve about a vertical diameter with uniform angular velocity: a small ring, capable of sliding upon the wire, would remain at rest relatively to the wire at a point of which the radius is inclined at an angle a to the vertical: to find the length of an isochronous simple pendulum for oscillations of the ring when slightly displaced from its position of relative rest.

If a be the radius of the circle, the length of the pendulum is equal to a cot a cosec a.

(12) A tube in the form of a cardioid, the axis of which is

equal to 2a, rotates with a uniform angular velocity (2)*

about

its axis, which is vertical, the cusp being at the top. A particle is projected within the tube at its lowest point with a velocity (3ga): to find the greatest altitude to which the particle will ascend.

The particle when at its greatest altitude is in a horizontal line with the cusp.

(13) A particle falls from rest towards a fixed centre of force, which attracts directly as the distance: to find the equation to the path of the particle, supposing it to be included in a thin smooth rectilinear tube, which passes through the centre of force and revolves in one plane with a uniform angular velocity.

=

=

Let the absolute force, w the angular velocity of the tube; and let a, the initial distance of the particle from the centre of force, be taken as the prime radius vector: then the equation to the path will be

according as μ is less or greater than w2.

If μ=w, the path becomes a circle.

(14) A particle P (fig. 165) is fixed to one end of a rigid rod PQ, which lies upon a smooth horizontal plane, and is so fine that its mass may be neglected. The end Q is constrained to move with a uniform velocity in the circumference of a circle ABQ; to find the velocity of the increase of the angle PQR, O being the centre of the circle, and OQR a straight line.

=

If PQ=h, OQ=a, PQR at any time t, a = the initial value of y, w = the angular velocity of OQ, B = the initial value

Clairaut; Mém. de l'Acad. des Sciences de Paris, 1736, p. 14.

(15) QBA (fig. 166) is a circle on a horizontal plane, and QP a string touching it at the point Q; P is a particle attached to the end of the string. Supposing the particle P to be projected at right angles to QP with a given velocity so as to cause QP to be gradually wrapped about the circumference QBA; to find the velocity of the particle at any time during the motion, and the time which will elapse before the particle reaches the circumference.

Let ẞ be the velocity of projection, v the velocity at any time during the motion, b the length of the string PQ, a the radius of the circle, T the time required. Then

(16) A circular horizontal lamina of matter ABC, (fig. 167), every particle of which attracts with a force varying inversely as the distance, is made to revolve with a uniform angular velocity round an axis through its centre O at right angles to its plane, the motion taking place in the direction of the arrows: to find the equation to the groove Aa which must be carved in the circular lamina in order that it may be described freely by a particle subject to the attraction of the lamina; the initial position of the particle being a point A in the circumference of the circle, and its initial velocity being zero.

Let P be any point in the groove; let OP=r, OA=a, ▲ POA = 0, ∞ = the angular velocity of the lamina about O, and f= the attraction of the lamina on a particle in its circumferThen the equation to the groove Aa will be

ence.

(17) Two small equal bodies A, B, connected together by a rigid line, are placed in a narrow rectilinear tube, in which they can move without friction; the tube is then made to revolve with a uniform angular velocity round a vertical axis which passes through a point C of the tube, this point C lying initially between A and B at a distance a from A and b from B: to find the time of A's arriving at C, and the tension of the rigid line at any time, a being considered less than b.

If a denote the angular velocity of the tube, m the mass of each particle, t the required time, and T the tension; then

(18) A particle is drawn up an indefinitely thin cycloidal tube, the axis of the cycloid being vertical, by means of an equal particle, to which the former particle is attached by a thread passing over a pully at the highest point of the arc to find the time of ascending to the highest point.

If T represent the required time, and t the time of a semioscillation in the cycloid,

T= 21t.

(19) A particle having been placed at a point in a straight line in a horizontal plane of indefinite extent, round which line as an axis the plane is then made to revolve with a uniform angular velocity: to find what time will elapse before the particle leaves the plane.

If o be the angular velocity and t the required time, then

This problem was proposed in the Lady's Diary, for the year 1778, by John Landen, by whom a solution was given, which is singularly defective, not only in consequence of his neglecting the consideration of centrifugal force, but also from his erroneously supposing the horizontal velocity of the particle to be equal to its velocity along the plane, multiplied by the cosine of the plane's inclination to the horizon. See Diarian Repository, p. 512, where a correct solution is given by the Editors of the Repository, together with Landen's.

SECT. 7. Constrained Motion of a Particle in Resisting Media.

(1) A particle descends down a straight line AB, (fig. 168), inclined at an angle a to the vertical, in a medium of uniform density, in which the resistance varies as the velocity to determine the velocity and the space at the end of any time.

=

Let P be the position of the particle at the end of any time t, vits velocity; let AP=x, and the resistance for a unit of velocity. Then, since the resolved part of the force of gravity along AB is at every point g cos a, we have for the motion of P,