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Ch. 13 - Fluids
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 13 - FluidsProblema 80a
Capitolo 13, Problema 80a

A 3.2-N force is applied to the plunger of a hypodermic needle. If the diameter of the plunger is 1.3 cm and that of the needle is 0.20 mm, with what force does the fluid leave the needle?

Guida verificata passo dopo passo
1
Step 1: Recognize that this problem involves Pascal's principle, which states that pressure applied to a confined fluid is transmitted undiminished throughout the fluid. The pressure at the plunger is equal to the pressure at the needle.
Step 2: Calculate the area of the plunger using the formula for the area of a circle: A = πr². Convert the diameter of the plunger (1.3 cm) to radius in meters (r = diameter/2), and then substitute into the formula.
Step 3: Calculate the pressure exerted by the plunger using the formula P = F/A, where F is the force applied to the plunger (3.2 N) and A is the area of the plunger calculated in Step 2.
Step 4: Calculate the area of the needle using the same formula for the area of a circle: A = πr². Convert the diameter of the needle (0.20 mm) to radius in meters (r = diameter/2), and then substitute into the formula.
Step 5: Use the fact that the pressure at the plunger equals the pressure at the needle (P_plunger = P_needle). Rearrange the pressure formula to solve for the force at the needle: F_needle = P_needle × A_needle. Substitute the pressure from Step 3 and the area of the needle from Step 4 to find the force with which the fluid leaves the needle.

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Pascal's Principle

Pascal's Principle states that when pressure is applied to a confined fluid, the pressure change is transmitted undiminished throughout the fluid. This principle is crucial for understanding how forces are transmitted in hydraulic systems, such as a hypodermic needle, where the force applied to the plunger results in a proportional force exerted on the fluid exiting the needle.
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Pressure Calculation

Pressure is defined as the force applied per unit area (P = F/A). In the context of the hypodermic needle, calculating the pressure exerted by the plunger allows us to determine how that pressure translates to the force exerted by the fluid as it exits the needle. The difference in cross-sectional areas between the plunger and the needle plays a critical role in this calculation.
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Area Ratio

The area ratio between the plunger and the needle is essential for determining the force exerted by the fluid. Since the force exerted by the fluid is related to the pressure and the area through which it flows, understanding how the areas of the plunger and needle compare allows us to calculate the force with which the fluid exits the needle, based on the initial force applied.
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