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Chapter 14: Problem 53

A hypodermic syringe contains a medicine with the density of water (Figure \(\mathbf{P} 1 \mathbf{1} . \mathbf{5 3}\) ). The barrel of the syringe has a cross sectional area \(A=2.50 \times 10^{-5} \mathrm{m}^{2},\) and the needle has a cross-sectional area \(a=1.00 \times 10^{-8} \mathrm{m}^{2} .\) In the absence of a force on the plunger, the pressure everywhere is 1 atm. \(A\) force \(\mathbf{F}\) of magnitude \(2.00 \mathrm{N}\) acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip.

Short Answer

Expert verified
The speed of the medicine as it leaves the needle is calculated using Bernoulli's principle and the continuity equation together. The resulting speed should be reported based on the calculated pressures and areas.

Step by step solution

01

Compute Initial Pressure in the Syringe

The initial pressure in the syringe is atmospheric pressure, given to be 1 atm. We'll need to convert this to pascals for consistency with our other units. 1 atm is equivalent to \(1.01 \times 10^{5} \) Pa.
02

Find Final Pressure in the Syringe

When the force is applied on the plunger, this creates an additional pressure in the fluid. Pressure is defined as force divided by the area over which the force is distributed. So, the additional pressure can be calculated using the formula \(P_{2} = F/A\), where \(F = 2.00 N\) is the force applied and \(A = 2.50 \times 10^{-5} m^2\) is the cross-sectional area of the barrel of the syringe. The total pressure in the syringe is the initial pressure plus this additional pressure.
03

Apply Bernoulli's Principle and Continuity Equation

Bernoulli's principle tells us that the fluid pressure decreases as the fluid's speed increases. That means the fluid pressure at the needle's tip is \(1.01 \times 10^{5} \) Pa (back to atmospheric pressure). By setting the total energy at the barrel (where the force is applied) equal to the total energy at the needle's tip (where the fluid leaves), we can apply Bernoulli's principle to find the speed of the medicine as it leaves. The continuity equation, which states that \(A_{1}v_{1} = A_{2}v_{2}\) (where \(v\) is velocity and \(A\) is cross-sectional area), tells us that the speed of the medicine is slower in the barrel than at the needle's tip due to the areas being different. This allows us to solve for \(v_{2}\), the speed of the medicine as it leaves the needle.
04

Compute Speed

Use the calculated pressures and areas along with the continuity equation to compute the speed of the medicine as it leaves the needle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Dynamics in Simple Terms
Think of fluid dynamics as the study of liquids and gases on the move. It's like understanding how water flows through a river or air blows through a vent. To get this, we use physics to see how fluids behave when they're pushed and pulled by various forces.

For our problem with the hypodermic syringe, we're interested in how the medicine, which flows like water, shoots out when a force is applied to the plunger. To solve this, we need to consider both pressure鈥攖he push on the fluid鈥攁nd the medicine's speed. Imagine squeezing a water balloon; where you press, it bulges out faster elsewhere. This is part of what fluid dynamics tackles, helping us predict the speed at which the medicine squirts out of a syringe.
Pressure Calculation Made Easy
You can think of pressure like the focus of a force over a certain area. It's kind of like wearing snowshoes to avoid sinking into the snow鈥攖he larger the shoe, the less you'll sink because your weight spreads out. To calculate the pressure in our syringe problem, we imagine the force of your thumb pushing down on the plunger. Divide this force by the area of the plunger's end, and you get pressure.

This additional pressure, on top of the atmospheric pressure already present, is what's trying to push the medicine out. Calculating this lets us figure out just how much extra push the medicine is getting in the syringe before it jets out of the needle.
Continuity Equation: A Fluid's Game of Telephone
The continuity equation in fluid dynamics is like a game of telephone鈥攚hat goes in must come out, but the message (or in our case, the fluid) might change along the way. It's a way to say that the amount of fluid flowing into a pipe has to equal the amount flowing out, assuming we're not adding or losing any fluid.

For the syringe, the equation tells us that the slow-moving medicine inside the bigger barrel has to speed up when it gets to the narrow needle to keep the flow consistent. Think of it as squeezing a tube of toothpaste鈥攖he paste comes out much faster at the tiny opening. This concept is key to finding out just how fast the medicine exits the needle when you apply a certain amount of force with the plunger.

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Most popular questions from this chapter

A Pilot tube can be used to determine the velocity of air flow by measuring the difference between the total pressure and the static pressure (Fig. \(\mathrm{P} 14.49) .\) If the fluid in the tube is mercury, density \(\rho_{\mathrm{Hg}}=13600 \mathrm{kg} / \mathrm{m}^{3},\) and \(\Delta h=5.00 \mathrm{cm},\) find the speed of air flow. (Assume that the air is stagnant at point \(A,\) and take \(\rho_{\mathrm{air}}=1.25 \mathrm{kg} / \mathrm{m}^{3} .\) )

Water flows through a fire hose of diameter \(6.85 \mathrm{cm}\) at a rate of \(0.0120 \mathrm{m}^{3} / \mathrm{s} .\) The fire hose ends in a nozzle of inner diameter \(2.20 \mathrm{cm} .\) What is the speed with which the water exits the norsle?

The water supply of a building is fed through a main pipe \(6.00 \mathrm{cm}\) in diameter. A \(2.00-\) cm-diameter faucet tap, located \(2.00 \mathrm{m}\) above the main pipe, is observed to fill a \(25.0-\mathrm{I}\) container in \(30.0 \mathrm{s}\). (a) What is the speed at which the water leaves the faucet? (b) What is the gauge pressure in the 6-cm main pipe? (Assume the faucet is the only "leak" in the building.)

The human brain and spinal cord are immersed in the cerebrospinal fluid. The Auid is normally continuous between the cranial and spinal cavities. It normally exerts a pressure of 100 to \(200 \mathrm{mm}\) of \(\mathrm{H}_{2} \mathrm{O}\) above the prevailing atmospheric pressure. In medical work pressures are often measured in units of millimeters of \(\mathrm{H}_{2} \mathrm{O}\) because body fluids, including the cercbrospinal fluid, typically have the same density as water. The pressure of the cerebrospinal fluid can be measured by means of a spinal tap, as illus. treated in Figure P14.21. A hollow tube is inserted into the spinal column, and the height to which the fluid rises is observed. If the fluid rises to a height of \(160 \mathrm{mm}\), we write its gauge pressure as \(160 \mathrm{mm} \mathrm{H}_{2} \mathrm{O} .\) (a) Express this pres. sure in pascals, in atmospheres, and in millimeters of mercury. (b) Sometimes it is necessary to determine if an accident victim has suffered a crushed vertebra that is blocking flow of the cerebrospinal fluid in the spinal column. In other cases a physician may suspect a tumor or other growth is blocking the spinal column and inhibiting flow of cerebrospinal fluid. Such conditions can be investigated by means of the Queckensied test. In this procedure, the veins in the patient's neck are compressed, to make the blood pressure rise in the brain. The increase in pressure in the blood vessels is transmitted to the cercbrospinal fluid. What should be the normal effect on the height of the fluid in the spinal tap? (c) Suppose that compressing the veins had no effect on the fluid level. What might account for this?

Water falls over a dam of height \(h\) with a mass flow rate of \(R,\) in units of \(\mathrm{kg} / \mathrm{s} .\) (a) Show that the power available from the water is $$\mathscr{P}=R g h$$ where \(g\) is the frec-fall acceleration. (b) Each hydroclectric unit at the Grand Coulce Dam takes in watcr at a rate of \(8.50 \times 10^{5} \mathrm{kg} / \mathrm{s}\) from a height of \(87.0 \mathrm{m} .\) The power developed by the falling water is converted to clectric power with an efficiency of \(85.0 \% .\) How much electric power is produced by each hydroelectric unit?
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