The Equivalent Mass of a Spring Vibrating LongiUidinally . 567 
E 
The acceleration of M is 
— Ap2 ijt . sin^Z ^ \ 
— MAj;2 sin pt . sin j^l /\/ ^ =M^-T,=:— E . 
:=-EAi>,>yp . sinp^ . cosK ^ ^ 
E ^ V E 
ie. Mp tan^Z V/ijE 
Let m be tlie wliole mass of the spring, and K the force necessary to 
produce unit extension of it, then 
p=z^ and E=:KZ 
tan pi j\J ^= VmK 
To deal with this equation write ^^^'Y^ ; the equation becomes 
M_cot « 
Let m' be the equivalent mass of m for the oscillation ; m is defined bv 
period=2,r 
, m r 1 cot ^"1 
•■• '"=g.-M = mL-^, ^-J 
§ 3. For particular values of M and m the equation for Q can be solved 
to any stated degree of accuracy and the corresponding value of m' found. 
The two extreme cases may be noted : 
(a) M small — 
Approximately ^—^ • * • ^2 
2 
This is very nearly ^ m,. 
(b) M great — 
Here 6^ = 0 m' ■ 
1 
3 6-' 
This is the result generally used. 
It is interesting to observe that the addition of a mass M to the end of 
