EXAMPLES XX 847 



(_' I) A body of mass m Ib. falls from rest in a resisting medium, 

 tin resistance of which is kv Ib., where v is the velocity of the 

 body and A; is a constant. If after the 1st second the velocity 

 i> 'JO ft. per sec. and after the 2nd second it is 85 ft. per sec., 

 show that the body after falling for a great length of time 

 will tend to have a velocity whose value is 111-9 ft. per sec. 

 (g = 82-2 f.s.s.). 



(25) In dealing with the strength of thick cylinders, if p is the 

 radial compressive stress and /the hoop tensile stress at a point 

 whose distance from the axis is r, 



Then p+f= 2a and p + r-~ =/ where a is a constant. If at 



the internal surface p = p Q when r = r and at the external sur- 

 face p = Pi when r = r lt express p in terms of r. 



(26) In a hollow cylinder of nickel steel subjected to internal 

 pressure p, and no pressure outside, when the material is all 

 yielding, if p is the radial compressive stress and / the hoop 

 tensile stress at a point whose distance from the axis is r, and 

 if / t- ap = b where a and b are constants for a particular kind 



of steel, and if we also have the usual relation r-~ 



dr 



find p as a function of r. If the inside radius r^ is 3 inches and 

 the inside p t is 30 tons per sq. inch, what is r , the outer radius. 



Q 



(Take for nickel steel a = -, b = 30.) (B. of E., 1911.) 



(27) A quantity of gas expands from 2-5 cubic ft. to 9 cubic ft., 

 the law of expansion being pv n = const. If the pressure at the 



beginning of the expansion is 80 Ib. per sq. inch, find the work 

 done during expansion : (1) when n = 1-0646, (2) when n = 1-131. 



(28) In the previous example, if the law of expansion is 

 pv = const. Find the work done during expansion. 



(29) Let p denote the population of England and Wales in 

 millions and t the time in years that has elapsed since 1801. If 

 the increase of population per year is proportional to the popu- 

 lation that is, it follows the compound interest law express p 

 in terms of t. If p was.8-9 in 1801, and it was 36-1 in 1911, what 

 is p likely to be in 1921 ? (B. of E., 1914.) 



(30) If y = Ac* 8 what is -^ ? An electric condenser, of capacity 



cue 



k farads and leakage resistance R ohms, has been charged and 

 the voltage v is diminishing according to the law -=- = -r=j. 



'// AAV 



Express v in terms of the time, t seconds. If k is 0-8 x 10~* 

 farads ; if v is noted to be 30 and 15 seconds afterwards it is noted 

 to be 26-43, find R. (B. of E., 1912.) 



