Question

Find the pressure increase in the fluid in a syringe when a nurse applies a force of \(42.3 \mathrm{~N}\) to the syringe's piston of diameter \(1.12 \mathrm{~cm} .\)

Short answer

The pressure increase in the fluid in the syringe when the nurse applies a force of \(42.3 \mathrm{~N}\) is \(4.29 \times 10^5 \mathrm{~Pa}\).

Step-by-step solution

Identify the given values

The force applied by the nurse is \(42.3 \mathrm{~N}\) and the diameter of the syringe's piston is \(1.12 \mathrm{~cm}\).

Convert diameter to radius and centimeters to meters

The formula for pressure involves the radius of the area, not the diameter. So, firstly, identify the radius, which is half of the diameter. Also, convert the radius from centimeters to meters for consistency in units. So, the radius, \(r = \frac{1.12 \mathrm{~cm}}{2} = 0.56 \mathrm{~cm} = 0.0056 \mathrm{~m}\).

Calculate the cross-sectional area of the piston

The cross-sectional area of the piston (A) is the area across which the force is being applied. This can be calculated using the formula for the area of a circle, \(A = \pi r^2\). Therefore, \(A = \pi (0.0056 \mathrm{~m})^2 = 9.85 \times 10^{-5} \mathrm{~m}^2\).

Use the pressure formula to calculate the pressure increase

Pressure (P) is defined as the force per unit area. Using this definition and the values calculated and identified earlier, we can write \(P = \frac{F}{A}\). Substituting the given and calculated values into this equation gives \(P = \frac{42.3 \mathrm{~N}}{9.85 \times 10^{-5} \mathrm{~m}^2} = 4.29 \times 10^5 \mathrm{~Pa}\), which represents the pressure increase in the fluid in the syringe.

Key concepts

Understanding the Pressure Formula in Physics

The concept of pressure is critical in various fields such as physics, engineering, and medicine. It describes the amount of force applied over a specific area. In physics, the pressure formula is given by \( P = \frac{F}{A} \) where \( P \) represents the pressure, \( F \) is the force applied, and \( A \) is the area over which the force is distributed.

Imagine pressing your finger against a balloon with a certain force. The smaller the area of contact, the more pressure you exert on that specific point, which increases the likelihood of the balloon popping. This principle is what makes sharp objects like needles and knives effective; they apply force over a very small area, resulting in high pressure.

When we talk about fluids, such as in the syringe exercise, escaping fluid applies pressure on all sides uniformly. Hence, when the nurse applies a force on the piston of the syringe, the pressure is transmitted equally in all directions within the incompressible fluid. The calculation of pressure is essential to ensure the administration of the correct amount of fluid in medical procedures.

Cross-Sectional Area of a Piston

Significance of the Piston's Cross-Sectional Area

In machines and devices like syringes, pistons play a pivotal role by converting force into mechanical motion or pressure. The cross-sectional area of a piston, typically circular in shape for syringes, is the surface over which the force is applied and is crucial for determining the pressure exerted by the piston.

To find this area, we use the formula for the area of a circle, \( A = \pi r^2 \) where \( A \) is the area, \( \pi \) (approximately 3.14159) is a constant representing the ratio of a circle's circumference to its diameter, and \( r \) is the radius of the piston. As you might recall from the exercise, the radius is half of the diameter, which in this case had to be converted from centimeters to meters to match the standard units used in physics for such calculations.

Understanding this concept helps us comprehend how different piston sizes affect the pressure exertion. Larger pistons with greater cross-sectional areas will distribute the same force over more space, leading to a reduction in pressure compared to smaller pistons.

The Relationship Between Force and Pressure

The relationship between force and pressure is straightforward yet significant: the greater the force applied to an object or fluid over a fixed area, the higher the pressure. This linear relationship is clearly portrayed in the pressure formula mentioned earlier.

However, if you apply the same force over a larger area, the resultant pressure decreases. Conversely, applying the same force over a smaller area increases the pressure. This concept is not just theoretical, but quite practical. It explains why sharp knives cut better than blunt ones (force is applied over a smaller area) and why wider tires can reduce the pressure on the ground to prevent sinking into soft surfaces.

In the syringe example, you can visualize the effect: by pushing down on the piston (a relatively small cross-sectional area), the nurse significantly amplifies the pressure exerted on the fluid inside. This relationship is crucial in accurately delivering medicines and drawing fluids from a patient's body.