How much more thumb pressure must a nurse use to administer an injection with a hypodermic needle of inside diameter 0.30 mm compared to one with inside diameter 0.60 mm? Assume that the two needles have the same length and that the volume flow rate is the same in both cases.

Explanation

The pressure required to achieve the same flow rate through two pipes with different diameters is given by the equation:P = 8Lnv/πr^4where P is the pressure, L is the length of the pipe, n is the viscosity of the fluid, v is the volume flow rate, and r is the radius of the pipe.Assuming the same length, viscosity, and flow rate for both needles, we can see that the pressure required is inversely proportional to the fourth power of the radius. That means the pressure required for the needle with the smaller inside diameter (0.30 mm) will be:P1 = 8Lnv/πr1^4And the pressure required for the needle with the larger inside diameter (0.60 mm) will be:P2 = 8Lnv/πr2^4Taking the ratio of the two pressures, we get:P1/P2 = (r2/r1)^4P1/P2 = (0.60/0.30)^4 = 2^4 = 16So the nurse would need to apply 16 times as much thumb pressure to achieve the same flow rate with the smaller needle.