Δ. A. 139. The pressure of the particle on the curve is – m§, the positive direction of § being towards the center of curvature. When the particle is merely on the curve, since the reaction can never be in the direction from the particle to the curve, whenever takes the value zero, the particle will leave the curve. If, however, the reaction be capable of being exerted in any direction, as in the case of a particle in a tube, there will be no such limitation. 140. To determine the motion of a heavy particle on a cycloid whose axis is vertical and vertex downwards. Let P be the position of the particle at any time, A the vertex of the cycloid, RPT the generating circle, whose diameter RT is vertical and = 2a. R T A Then PT is a tangent to the cycloid (Appendix, Art. 2). this varies as the distance from A. The other part of the force and the constraint are only concerned with changing the direction of the motion, i. e. keeping the particle on the curve. This kind of motion has been investigated in Arts. 51-54, and in Newton, § ij. The particle will oscillate between two points equally dis tant from A, the time of a complete oscillation being 2π 4a The velocity at any point is determined from the considerations in Art. 133. A. 141. This may be easily got from the equations to the cycloid cos 0), y=a (0+ sin 0); (App. Art. 5), dx ds where the axis of x is measured vertically upwards. d's Then = 9 is the equation of motion. Therefore from Arts. 51-54, the motion is oscillatory and the time of a complete oscillation is 2π √ 4a 142. It must be observed that in this case of motion the time of one oscillation is independent of the initial distance, or the oscillations in a cycloid are isochronous. 143. By means of two vertical semi-cycloidal cheeks connected so as to form a cusp, a particle suspended by a string can be made to oscillate in a cycloid, as indicated in App. Art. 4: the string will manifestly be always stretched. This forms a "simple pendulum." The length of this string is 4a=l say, and therefore the time of a complete oscillation = 2π In pendulums the time of an oscillation is generally taken to be the time from rest to rest, and therefore it = π This formula is made great use of for finding the value of g: for the length 7 of a seconds' pendulum can be determined with great accuracy, and then we have 1 =π whence g: = T2l: * The manner in which this is ascertained will be found in Griffin's Dynamics of a Rigid Body. L the units in which g is expressed are seconds, and whatever unit of length is assumed in 7. The length of a seconds pendulum = 39.1392 inches, very nearly, i. e. 3.2616 feet; and putting π=3·1416, we have g= 32.19 feet per second, nearly. 144. To determine the motion of a heavy particle in a vertical circle. If the motion is very small, it will consist of small oscillations about the lowest point, which can be deduced from the case of the cycloid, the time of one of these complete oscillations being 2π where is the radius of the circle. In other cases we must proceed as follows: Let C be the initial position of the particle, Pits position at any time, A the lowest point of the circle, O its center, a = its radius, v, v' = the velocities at P, C respectively, angle AOP 0, AOC= a; and CM, PN horizontal lines. tical Then the change of velocity from C to P is due to the verspace descended (Art. 133); .. v2 — v'2 = 2g. MN=2ga (cos 0 — cos a). Let be the acceleration due to the force of constraint in the normal, considered positive if towards O. = acceleration in PO= -g cos 0 (NEWTON, § ij); therefore if R be the force of constraint, and W the weight of If the particle be on the outside of the circle, R must never be positive; if on the inside, or if suspended from O by a string, R must never be negative; but if the particle be moving in a circular tube or groove, or be connected with O by a rigid rod, R may be either positive or negative. 145. If the particle be initially at rest, v' = 0; and then whence cose > cos a, or lies between +a and -a; therefore the motion is oscillatory between the points for which 0=+a and α. The greatest value of R in any case is when cos is greatest, i.e. when 0=0, or at the lowest point. If the particle start from rest at the highest point, a =π, and the greatest value of R is 5 W. 146. If the particle be suspended by a string, making complete revolutions, and if the velocity at the highest point be just sufficient to keep the string stretched, And we have .. v=ga. The greatest value of this is 6 W: i. e. the string must be able to bear six times the weight of the particle without breaking, in order that this motion may continue. 147. The motion of Art. 145 will not apply to a particle suspended by a string unless R is always positive: the least value of R corresponds to the least value of cos 0, i. e. when 0 = =a, and R then becomes W cos a. If this is positive, a must π not exceed : therefore oscillatory motion of a particle sus 2 pended by a string is not possible unless the extent of oscillation be not greater than a semicircle. A. 148. All this might have been obtained by means of the equations in Arts. 34 or 35; which, remembering that in a circle is constant and = a, and s= a0, become γ If be so small that its cube and higher powers may be neglected, the second equation becomes whence, by Arts. 51-54, the motion will be oscillatory, and the time of a complete oscillation will be 2π a therefore a pen dulum oscillating in a circle may be considered as one oscillating in a cycloid, when the arc of vibration is so small that its cube and higher powers may be neglected. |