Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 29b

Chapter 14, Problem 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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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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. What is the minimum force the nurse needs to apply to the syringe?

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