Ahypodermic needle consists of a plunger of circular cross-section that slides inside a hollow cylindrical syringe. when the plunger is pushed, the contents of the syringe are forced through a hollow needle (also of circular cross-section). if a 4.0-n force is applied to the plunger and the diameters of the plunger and the needle are 1.2 cm and 2.5 mm, respectively, what force, in n, is needed to prevent fluid flow at the needle?

The fluid that is being passed through the syringe and needle is incompressible, which means that it will transmit pressure equally. Therefore, the pressure on the plunger will be equivalent to the pressure on the needle. We also know that:

Pressure = Force / Area

Pressure on plunger = 4 / (π*(0.012/2)²)
Pressure on plunger = 35.4 kPa
Pressure on needle = 35.4 kPa

35.4 kPa = F / (4 / (π*(0.0025/2)²)
F = 0.17 N

The force on the needle is 0.17 N