162
CONTACT OF ELASTIC SOLIDS
1
de
= 1.4716.
Thus the time of impact may become infinite in various ways
without the time, with which it is to be compared, also
becoming infinite. In particular the time of impact becomes
infinite when the initial relative velocity of the impinging
bodies is infinitely small; so that whatever be the other
circumstances of any given impact, provided the velocities
are chosen small enough, the given developments will have
any accuracy desired. In every case this accuracy will be
the same as that of the so-called laws of impact of perfectly
elastic bodies for the given case. For the direct impact of
two spheres of equal radius R and of the same material of
density q the constants k₁ and 2 are
=
3
2R$πq
K₂
=
8 /R
3k
hence in the particular case of two equal steel spheres of
radius R, taking the millimetre as unit of length, and the
weight of one kilogramme as unit of force, we have
log k₁ = 878-3 log R,
log k₁ = 4·03 + log R.
Thus for two such spheres impinging with relative velocity v:
the radius of the surface of impact
the time of impact
•
the total pressure at the instant of
nearest approach
·
•
the simultaneous maximum pressure
at the centre of impact per unit
area
=
am=0·0020Rv³mm,
T 0.000024 Rv *sec,
=
Pm = 0.00025R²v¹kg,
P'm 29.1 kg/mm².
=
For instance, when the radius of the spheres is 25 mm., the
velocity 10 mm/sec, then a 0.13 mm., T=0.00038 sec.,
Pm = 2·47 kg., P'm = 730 kg/mm.² For two steel spheres as
large as the earth, impinging with an initial velocity of 10
mm/sec, the duration of contact would be nearly 27 hours.