MEASUREMENT OF PRESSURES 153 



difference that the piston and crusher are initially a known distance 

 from the anvil rather than in contact with it, as indicated diagram- 

 matically in Fig. 5.3(a). The piston thus has a length L of free travel 

 before striking the crusher and in this time acquires a momentum pro- 

 portional to the impulse or time integral of the pressure on its face. 

 The impulse per unit area up to the time t at which the piston strikes 

 the crusher will be denoted by lit), where 



(5.6) m = fp{t) 



dt 



We may write, by conservation of momentum, that the velocity V of 

 the piston on impact is given by 



(5.7) ■ lit) = ^ 



where m is the effective mass of the piston and A the area exposed to 

 pressure. Hilliar found, however, that the deformation S of the crusher 

 as a result of impact by a piston is a function only of the energy of the 

 piston. With the aid of calibration tests using pistons of known mass 

 shot at measured velocities from a compressed air "gun," a calibration 

 curve of kinetic energy Ec absorbed by the crusher versus crusher de- 

 formation can be prepared. If the deformation of a crusher struck by 

 a piston accelerated by the pressure wave is then measured, its value 

 gives a measure of the energy Ep oi the piston and hence its velocity 

 V{t). Two corrections are necessary: the body of the gauge is also 

 moving, although more slowly, and the piston does work on the copper 

 before being brought to rest because of the pressure still acting. The 

 first of these effects is accounted for by using a reduced mass 

 ffi' = m/{l + m/M) where M is the mass of the gauge, and the second 

 by subtracting a term P • S from the calculated energy absorbed, where 

 P is the pressure during impact, with the result 



^'■'^ ^^ = TTWm^^'-'''^ 



The kinetic energy of the piston is, however, given by ^p = ^mV^ and 

 substituting in Eq. (5.7) gives 



(^•9) ^« = jVrTS7M^^'-^^) 



