CHAPTER V 



COEFFICIENTS OF INERTIA 

 146. EFFECTS OF FLUID INERTIA. 



When a solid body submerged in an incompressible nonviscous fluid is acted upon by 

 an external force, an acceleration is produced which is less than that for the same body in a 

 vacuum, since the external force must also accelerate the fluid surrounding the body. It will 

 be shown that the effect of the fluid can be represented by assigning to the solid an equiva- 

 lent mass greater than the mass of the solid itself. 



If the motion of the body is one of translation at velocity C/ in a fixed direction while 

 the fluid is at rest at infinity, the velocity of the fluid at every point will be proportional to 

 U and its kinetic energy will be proportional to f/^. Hence, the total kinetic energy of body 

 and fluid can be expressed in the form 



T = - (M + kM') i/2 [146a] 



where M is the mass of the body, 



M' is the mass of fluid displaced by it, and 



k is a constant of proportionality called the coefficient of inertia. 

 The value of k will depend upon the size and shape of the body and, in general, upon its orien- 

 tation relative to the direction of motion. 



If F is the external force acting on the body in the direction of the motion, the rate at 

 which F does work must be equal to the rate of increase of the total kinetic energy; 



dT dU 



hence FU = — =(M + kM')U 



dt ' dt 



and F--(M + kM') [146b] 



dt 



Thus the acceleration produced is the same as if the mass of the body were increased from 

 M to M + kM', The added term kM' ma.)/ be considered as an effective mass due to the pres- 

 ence of the fluid. 



If U is constant, f = 0, so that no force is required to keep a body in motion provided 

 there is no fluid friction. The coefficient of inertia has significance in cases of acceleration 

 only. 



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