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Ch 14: Fluids and Elasticity
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 14: Fluids and ElasticityProblema 29b
Capitolo 14, Problema 29b

A 2.0 mL syringe has an inner diameter of 6.0 mm, a needle inner diameter of 0.25 mm, and a plunger pad diameter (where you place your finger) of 1.2 cm. A nurse uses the syringe to inject medicine into a patient whose blood pressure is 140/100. The nurse empties the syringe in 2.0 s. What is the flow speed of the medicine through the needle?

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Step 1: Calculate the volume flow rate of the medicine. The syringe is emptied in 2.0 seconds, and its volume is 2.0 mL. Convert the volume to cubic meters (m³) using the conversion factor: 1 mL = 1 × 10⁻⁶ m³. Then, divide the volume by the time to find the flow rate: \( Q = \frac{V}{t} \).
Step 2: Determine the cross-sectional area of the needle. The inner diameter of the needle is 0.25 mm. Convert this diameter to meters (m) and use the formula for the area of a circle: \( A = \pi \left( \frac{d}{2} \right)^2 \), where \( d \) is the diameter.
Step 3: Relate the flow rate to the flow speed using the continuity equation. The flow rate \( Q \) is equal to the product of the cross-sectional area \( A \) and the flow speed \( v \): \( Q = A \cdot v \). Rearrange this equation to solve for the flow speed: \( v = \frac{Q}{A} \).
Step 4: Substitute the values for \( Q \) (from Step 1) and \( A \) (from Step 2) into the equation for \( v \) to calculate the flow speed of the medicine through the needle.
Step 5: Ensure all units are consistent throughout the calculation (e.g., meters, seconds) and verify the result conceptually by checking if the flow speed is reasonable given the dimensions and time provided.

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Flow Rate

Flow rate is the volume of fluid that passes through a given surface per unit time. In this context, it can be calculated using the volume of the syringe and the time taken to empty it. The flow rate is crucial for determining how quickly the medicine is delivered through the needle.
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Fluid Speed & Volume Flow Rate

Continuity Equation

The continuity equation states that for an incompressible fluid, the product of the cross-sectional area and flow speed must remain constant along a streamline. This principle allows us to relate the flow speed in the syringe to the flow speed in the needle, given their different diameters.
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Cross-Sectional Area

The cross-sectional area of a cylindrical object, like a syringe or needle, is calculated using the formula A = π(d/2)², where d is the diameter. This area is essential for determining the flow speed, as it influences how much fluid can pass through at any given time, impacting the overall flow dynamics.
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