348 PRACTICAL MATHEMATICS 



(31) In the atmosphere, if p is pressure and h height above 

 datum level, if w = cp l h where c and y are constants, and if 



~ = w, find an equation connecting p and h. What is the 



cZ/t- 



above c if p = twR ? Assume p = p and t = t where h = 0. 

 R is a known constant for air. Find an equation connecting 

 h and t. (B. of E., 1904.) 



(32) Water leaves a circular basin very slowly by a hole at the 

 bottom, every particle describing a spiral which is very nearly 

 circular. Let v be the speed at a point whose distance from the 

 axis is r and height above some datum level h. Assume no 



" rotation " or " spin "that is, -(- + ) = and show that 



2 \r drJ 







this means v = - where c is some constant. Now at the atmo- 

 r 



v 2 

 spheric surface - + h = C where C is a constant. Find from 



2 



this the shape of the surface that is, the law connecting r and h. 

 (B. of E., 1905.) 



(33) If v volts is the voltage in an electric circuit, C amperes 

 the current, R ohms the resistance, L henries the self-inductance, 



jr\ 



and t seconds the time, then v = RC + L :- 



at 



If v is constant and equal to 8 volts, R = 075 ohms, and 

 L = 0-08 henry, express C in terms of t, knowing that when 

 t = 0, C = 0. What would be the value of C when t = 0-1 sec. ? 



(34) If v volts is the voltage in an electric circuit, C amperes 

 the current, R ohms the resistance, L henries the self-inductance, 



7/~( 



and t seconds the time, then v = RC + L r 



at 



If v = v sin pt where v and p are constants, and if R = 50 

 ohms, L = 0-1 henry, v = 100 volts, and p = 500, express C in 

 terms of t, knowing that when t = 0, C = 0. To what value does 

 C ultimately tend if t is taken sufficiently great ? 



