268 MOTION UNDER A VARIABLE FORCE 



simple harmonic, and consequently the period is different from that 

 of the simple harmonic motion described when the amplitude is 

 very small. Thus the period depends on the amplitude, so that a 

 clock which beats true seconds when the pendulum swings through 

 one angle will gain or lose as .soon as the pendulum is made to 

 swing through any different angle. Variations of amplitude must 

 always occur during the motion of any pendulum, and these cause 

 irregularities in the timekeeping of the clock. 



We have, however, seen that if a particle describes a cycloid, 

 the period is independent of the amplitude, so that variations of 



amplitude cannot affect the 

 timekeeping powers of a par- 

 ticle moving in a cycloid. 



The simplest way of caus- 

 ing a particle to move in a 

 cycloid is, in practice, to sus- 

 pend it from a fixed point by 

 a string, in such a way that 

 FIG 133 during its motion the string 



wraps and unwraps itself 



about two vertical cheeks. If the curve of these cheeks is rightly 

 chosen, the particle can be made to describe a cycloid, and it can 

 easily be shown that the curves o the cheeks must be portions of 

 two cycloids each equal to the cycloid which is to be described by 

 the particle. 



EXAMPLES 



1. In cycloidal motion prove that the vertical component of the velocity of 

 the particle is greatest when it has described half of its vertical descent. 



2. A particle oscillates in a cycloid under gravity, the amplitude of the 

 motion being 6 and the period being r. Show that its velocity at a time t meas- 



. 2irb . 2-n-t 

 ured from a position of rest is sin 



T T. 



3. A particle of mass ra slides on a smooth cycloid, starting from the cusp. 

 Show that the pressure at any point is 2 mg cos ^, where \f/ is the inclination to 

 the horizontal of the direction of the particle's motion. 



