Answer :
The pressure p(r)p(r)p(r) of the fluid in the test tube at an arbitrary distance rrr from the axis of rotation is p(r) = ρω²/2 (r²-ro²) + Po
The following figure shows the test tube filled with water spinning with angular velocity ω
The centripetal force acting on the particle of mass dm at a distance x is, F = dmω²x
The mass dm is given as,
dm = ρdv
= p Adx
Therefore, the force is,
F = ρAω²x dx
The pressure is force per unit area. Therefore, the pressure can be determined as follows:
P(r) = ρω²[tex][\frac{x^{2} }{2} ]^{r} _{r0} +Po[/tex]
Hence, the pressure is,
p(r) = ρω²/2 (r²-ro²) + Po
1 based on or influenced by one's own whims, biases, etc.; capricious. 2 not being absolute; merely having a relative use or importance. 3 tyrannical or dictatorial (of a government, ruler, etc.). 4 (Math) does not represent any particular value.
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