A 2.0 mL syringe has an inner diameter of 6.0 mm, a needle...

A $2.0 \mathrm{mL}$ syringe has an inner diameter of $6.0 \mathrm{mm},$ a needle inner diameter of $0.25 \mathrm{mm},$ and a plunger pad diameter (where you place your finger) of $1.2 \mathrm{cm} .$ A nurse uses the syringe to inject medicine into a patient whose blood pressure is $140 / 100$. Assume the liquid is an ideal fluid. a. What is the minimum force the nurse needs to apply to the syringe? b. The nurse empties the syringe in 2.0 s. What is the flow speed of the medicine through the needle?

A $2.0 \mathrm{mL}$ syringe has an inner diameter of $6.0 \mathrm{mm},$ a needle inner diameter of $0.25 \mathrm{mm},$ and a plunger pad diameter (where you place your finger) of $1.2 \mathrm{cm} .$ A nurse uses the syringe to inject medicine into a patient whose blood pressure is $140 / 100$. Assume the liquid is an ideal fluid.
a. What is the minimum force the nurse needs to apply to the syringe?
b. The nurse empties the syringe in 2.0 s. What is the flow speed of the medicine through the needle?

Key Concepts

Pascal's Principle

Pascal's Principle states that pressure applied to an incompressible fluid in a closed system is transmitted equally in all directions throughout the fluid. This concept is essential when relating the force applied on one part of the system (for example, the plunger) to the forces exerted elsewhere (such as at the needle), considering the differences in areas over which the pressure acts.

Pressure-Force-Area Relationship

The relationship between pressure, force, and area is fundamental in fluid mechanics, described by the equation P = F/A. This means that force is the product of pressure and the area over which it is applied. Understanding this principle allows one to determine the necessary force to achieve a given pressure when the areas of application differ, as in the case of a syringe and its needle.

Continuity Equation

The continuity equation for fluids states that the mass flow rate must remain constant for an incompressible fluid as it flows through varying cross-sectional areas. This implies that the velocity of the fluid is inversely proportional to the cross-sectional area of the channel it flows through. This principle is key for determining the flow speed through a narrow needle when the volumetric flow rate is known.

Ideal Fluid Assumption

Assuming an ideal fluid simplifies analysis by neglecting viscosity, turbulence, and other dissipative effects. This assumption allows one to apply principles like Pascal's Principle and the continuity equation directly without needing to account for energy losses, making it easier to compute forces and flow speeds in theoretical scenarios.

Transcript

Please give Ace some feedback