A hypodermic syringe contains a medicine with the density of...

A hypodermic syringe contains a medicine with the density of water (Fig. $\mathrm{P} 9.47$ ). The barrel of the syringe has a cross-sectional area of $2.50 \times 10^{-5} \mathrm{~m}^{2}$. In the absence of a force on the plunger, the pressure everywhere is $1.00 \mathrm{~atm} . \mathrm{A}$ force $\overrightarrow{\mathbf{F}}$ of magnitude $2.00 \mathrm{~N}$ is exerted on the plunger, making medicine squirt from the needle. Determine the medicine's flow speed through the needle. Assume the pressure in the needle remains equal to $1.00 \mathrm{~atm}$ and that the syringe is horizontal.

Key Concepts

Relationship Between Pressure and Flow Velocity

When a pressure differential is established in a fluid, some of the pressure energy is converted into kinetic energy, leading to an increase in flow speed. The quantitative relationship arising from energy conservation principles allows one to express the flow velocity as a function of the pressure drop and the fluid's density, forming a cornerstone of fluid dynamics analysis.

Force, Area, and Pressure Generation

The generation of pressure within a confined fluid system is governed by the relationship between force and cross-sectional area. By applying a force over a given area, one can induce a specific pressure level which, under the right conditions, accelerates the fluid and dictates its subsequent behavior within the system.

Bernoulli's Principle

This principle states that for an incompressible, non-viscous fluid, the sum of its pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline. It enables the conversion between pressure differences and fluid velocities, providing the basis for calculating flow speeds in systems where pressure boundaries differ.

Pressure Difference and Its Effects

A pressure difference in a fluid system creates a net force that drives the fluid from regions of higher pressure to regions of lower pressure. In practical applications, this difference can be produced by an external force acting over an area, and it is the starting point for generating fluid motion in devices such as syringes or pipelines.

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