12 PRACTICAL MATHEMATICS 



If the crank is making n revolutions per second, the angle 

 turned through per second is 27m radians, and if 6 is the angle 

 turned through in t seconds 



Then = limt radians 



r a 



and x = r(l cos 2-jrnt) -\ j (1 cos 4<7rnt) 



4t 



(2) The effect of heat on the pendulum. 



The time of swing of a pendulum is given by ir^J- where 



O 



I is the length of the pendulum in feet. 



Let /j be the length of a pendulum making n x beats per second 

 at a temperature of T x C. 



t 1 =7r/v/- 1 



Then t =7r/v- 1 and nt = 1 



O 



Let Z 2 be the length of the same pendulum at a temperature of 

 T 2 C. and at this temperature it makes n z beats per second. 



Then ^ ==7r '~ and n/ = 1 



but 1 2 = Zj{l + <x(T 2 T x )} where a is the coefficient of ex- 

 pansion of the pendulum rod. 



Now 



* 2 Vl +a(T 2 -T 1 ) 

 and ^ = (1 + a(T 2 - T\) }~* 



w oc 



Then 2 = 1 -(T 2 T x ) to the first approximation and 



n i 



' 2 T x ) + - (T 2 T x ) 2 to the second approxi- 

 mation, since a(T 2 T x ) is small compared with 1. 

 Working to the first approximation 



a IT T \ i n a _ n i ~ n 2 



o^ A 2 x l/ * "" 



Then w 1 n 2 = ^^ia(T 2 TJ giving the loss of the pendulum 



per second due to the temperature increasing from Tj to T 2 . 



If the range of temperature is great, it might be necessary to 

 take the second approximation. 



Then - (T T)- (T -T) 2 =l-^= MI ~ n<i 

 2 8 2 Wj w x 



f O-.2 "\ 



?5T1/I *W yvi /I 



ell Id /t"| ~~ /t-0 ' 



