Problem 47 - College Physics

Question

A hypodermic syringe contains a medicine with the density of water (Fig. P9.47). 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 \(1.00 \mathrm{~atm}\). 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.

Step-by-Step Solution

Answer

The flow speed of the medicine is \(13.4 m/s\)

Step 1: Identify key variables

The cross-sectional area of the syringe barrel \(A = 2.50 \times 10^{-5} m^{2}\), the force exerted on the plunger \(F = 2.00 N\), and the water's density \(\rho = 1000 kg/m^{3}\). Pressure is equal to 1.00 atm, which is \(P = 1.01 \times 10^{5} N/m^{2}\) when converted.

Step 2: Calculate the pressure due to the force exerted

Pressure \(P_{f}\) can be calculated using the equation \(P = F/A\), where \(F\) is the applied force and \(A\) is the cross-sectional area. Substituting the given values into the equation gives us \(P_{f} = 2.00 N / 2.50 \times 10^{-5} m^{2} = 80000 N/m^{2}\)

Step 3: Calculate the total pressure in the fluid

The total pressure \(P_{t}\) is the sum of the atmospheric pressure \(P\) and the pressure due to the applied force \(Pf\). Hence, \(P_{t} = P + P_{f} = 1.01 \times 10^{5} N/m^{2} + 80000 N/m^{2} = 1.89 \times 10^{5} N/m^{2}\)

Step 4: Calculate the speed of the fluid

We can use Bernoulli's equation to calculate the fluid speed, \(V = \sqrt{{2(Pt - P)}/{\rho}}\). Substituting the values into the formula gives us \(V = \sqrt{{2(1.89 \times 10^{5} N/m^{2} - 1.01 \times 10^{5} N/m^{2})}/{1000 kg/m^{3}}\) = 13.4 m/s

Key Concepts

Fluid Dynamics

Fluid dynamics is the study of how fluids move and the forces acting on them. It's a part of the broader field of fluid mechanics. Understanding fluid dynamics is crucial in many areas such as engineering, meteorology, and medicine. For this exercise, we focus on how a fluid behaves inside a hypodermic syringe. The fluid dynamics principles help us determine how pressure and flow speed change when we apply a force to the syringe plunger.

Bernoulli's equation is a key tool in fluid dynamics that relates pressure, flow speed, and height in a flowing fluid. It helps solve many real-world problems involving fluid flow, like calculating the flow speed in a syringe, as illustrated in this exercise.

Pressure Calculation

Pressure is the force exerted per unit area, and it's vital in understanding how fluids behave under different conditions. In our syringe example, the pressure is initially equal to the atmospheric pressure, 1.00 atm, or 1.01 x 10^5 N/m^2.

When a force is applied to the plunger, it creates additional pressure, calculated using the formula:

• \[ P_f = \frac{F}{A} \]

where \( F \) is the applied force and \( A \) is the cross-sectional area. By knowing these values, we calculate an additional pressure of 80000 N/m^2.

This additional pressure combines with atmospheric pressure to give the total pressure inside the syringe. Pressure calculations are fundamental in predicting how fluids move in various situations.

Cross-Sectional Area

The cross-sectional area of a tube or a syringe barrel determines how a fluid flows through it. In our exercise, this area is given as \( 2.50 \times 10^{-5} \text{ m}^2 \).

The cross-sectional area is crucial because it affects the pressure applied by the force on the plunger. A smaller area results in higher pressure, influencing how fast the fluid travels through the needle.

Understanding the concept of cross-sectional area helps in comprehending how variations in a tube's width can impact fluid flow, a principle used in designing everything from medical syringes to water pipes.

Hypodermic Syringe

A hypodermic syringe is a medical tool used to inject or withdraw fluids. It consists of a barrel, a plunger, and often a needle. In our exercise, we're using it as a model to understand fluid dynamics.

In this context, the syringe provides a practical example of how applied force, pressure, and fluid flow can be calculated. By applying a force to the plunger, we can manipulate how the medicine inside the syringe moves through the needle. Understanding the behavior of fluids within such devices is essential for clinical applications and designing medical equipment.

Flow Speed

Flow speed is the rate at which fluid moves through a particular point. In our problem, we determine the speed at which the medicine exits the needle using Bernoulli's equation.

By knowing the total pressure inside the syringe and the fluid density, we calculate the flow speed as follows:

• \[ V = \sqrt{\frac{2(P_t - P)}{\rho}} \]

This formula helps determine that the medicine flows at a speed of 13.4 m/s.

Understanding flow speed is important in various fields, as it impacts how fluids are transported efficiently in pipes or medical equipment.

Physics Problem Solving

Physics problem solving involves breaking down a problem into manageable steps and using appropriate formulas. In this exercise, we tackled the syringe problem step by step.

First, we identified the key variables: applied force, area, and atmospheric pressure. Next, we calculated pressures, applied Bernoulli's principle, and determined the flow speed.

Problem solving in physics often requires a systematic approach:

• Identify given information.
• Use relevant equations.
• Calculate step-by-step, checking units and conversions.

This structured approach helps dissect complex scenarios, making physics problems easier to understand and solve.