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SEP 2 61916 V) 
THE EQUIVALENT MASS OF A SPRINO VIBRATING 
LONGITUDINALLY. 
By Alexander Brown. 
(From the Applied Mathematics Laboratory, South African College.) 
(Received April 16, 1915.) 
§ 1. When a mass M is oscillating under gravity at the end of a spiral 
spring, it is usual to make allowance for the mass m of the spring itself by 
adding a quantity ^ m to M and treating the spring as if it were light. This 
result is correct only if m is small compared to M ; and in this case it is 
possible to give an elementary solution by supposing the displacement of 
any point on the spring to be proportional to its distance from the fixed end. 
In ordinary laboratory practice M is comparable with m in magnitude, and 
the above approximation no longer holds ; for small values of M it is 
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found that ^ 7n is a better value for the equivalent mass of the spring than 
^ m. It seems worth while to determine how this quantity changes as M 
changes. Rayleigh, in his book on Sound, vol. i, §§ 155-6, works out certain 
results in the longitudinal vibrations of bars which can be applied to this 
problem. The effect of a very small mass and of a very large mass added to 
a bar vibrating longitudinally are there shown ; and from them the results 
4 ^1 
— m and. - ^ can be deduced for the equivalent mass of m when M is very 
small and very large respectively. For moderate values of M the equivalent 
mass varies between the extremes mentioned, and for any actual m it is 
important to know the effect of the mass of the spring more closely. 
§ 2. We assume that the spring behaves like a uniform thin elastic cord. 
Let E be its elastic coefficient, fj its line density, and I its length ; M is the 
mass attached to the free end. 
Let £ be the distance of a particular element of the spring from the 
fixed end at any time ; x the distance when the spring is unstretched 
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