The Dissection of Rectilineal Figures (Concluded)

W. H. Macaulay
1915 Mathematical Gazette  
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more » ... GS OF A CARRIAGE BODY. 109 MOTION ON THE SPRINGS OF A CARRIAGE BODY. 109 It is usual in a mathematical investigation of damping to assume that the friction in a small oscillation is represented by a term proportional to the relative velocity of slipping, as this assumption makes the analysis tractable, and it gives a good representation of the facts, provided the relative velocity is low. The damped vibration is then represented by a periodic term of which the amplitude is qualified by a factor decreasing at compound discount, and Naperian logarithmic decrement is the name given to one-hundredth of the rate per cent per period of vibration at which the amplitude of vibration diminishes at continuous compound discount. In a graphical representation the effect is to change the circle in Fig. 2 , p. 392, Math. Gazette, into an equiangular spiral, as explained in Maxwell's Electricity and Magnetism, ? 731. 20. Consider a symmetrical motor car, where the four wheels are equal for interchangeability, and a,=a2= a, going slowly over the crest of a road of two equal inclines sloping at angle a, and meeting in a ridge, say over an old canal bridge. As the front wheels pass over the ridge, the c.G. of the car will proceed to describe an ellipse, with horizontal and vertical axis 2h + a cot a, 2ha tan a. Thus, if h= la tan a, the C.G. will advance in a horizontal line for a distance a(tana+cota)=2acosec2a; and if /e < a tana, the car can remain on the crest in stable equilibrium, and requires to be drawn off. Conversely at a dip in the road. Going fast up the incline at full speed S m.p.h., the front wheels would leave the ground at the ridge; and treating the car as a particle, it would proceed to describe a parabola, and lose contact with the road for a length 0'09S2 sin a yards, and for 0'18Ssin a seconds, and come down again at one in ? (cot a +3 tan a) with the ground. Up and down one in 20, for instance, at 30 m.p.h., the car will leap 1'8 yards in 0'18 second, and come down on the ground at one in 10. G. GREENHILL.
doi:10.2307/3603211 fatcat:x5uaklfzvfb6tgy5utuu32qvt4