Problem 3 - Fundamentals of Physics

Question

Find the pressure increase in the fluid in a syringe when a nurse applies a force of \(42 \mathrm{~N}\) to the syringe's circular piston, which has a radius of \(1.1 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The pressure increase in the fluid is approximately 110,497.37 Pa.

Step 1: Understand the Pressure Formula

Pressure is defined as force applied per unit area. The formula is given by:\[ P = \frac{F}{A} \]where \( P \) is the pressure, \( F \) is the force applied, and \( A \) is the area over which the force is applied.

Step 2: Calculate the Area of the Piston

The area of the circular piston can be calculated using the formula for the area of a circle:\[ A = \pi r^2 \]Substitute the radius of the piston \( r = 1.1 \text{ cm} = 0.011 \text{ m} \) to find the area:\[ A = \pi \times (0.011)^2 \approx 3.801 \times 10^{-4} \text{ m}^2 \]

Step 3: Calculate the Pressure Increase

Substitute the given force and the computed area into the formula for pressure:\[ P = \frac{F}{A} = \frac{42 \text{ N}}{3.801 \times 10^{-4} \text{ m}^2} \]Solve for \( P \):\[ P \approx 110,497.37 \text{ Pa} \]

Key Concepts

Force

In physics, force is what we use to describe any interaction that, when unopposed, changes the motion of an object. Think of it like a push or pull that makes something move or stop. The standard unit to measure force is the Newton (N). One Newton is the force required to accelerate a one-kilogram mass by one meter per second squared.

When a nurse pushes down on a syringe's piston, they're applying a force. That force is transferred to the liquid inside, increasing its pressure. In our exercise, the nurse applies a force of 42 Newtons to press down the syringe's piston.

Area of a Circle

The area of a circle is the space contained within its boundary, or the measurement of its surface. To find this, you use the formula:

• \( A = \pi r^2 \)

where \( A \) is the area and \( r \) is the radius of the circle.
Assume you have a circle with a radius of \( 1.1 \) cm, like the syringe's piston in our problem. First, you need to convert centimeters to meters, as the standard unit for calculating area in physics is square meters.

So, \( r = 0.011 \) m, and when you plug it into the formula, you find the area is approximately \( 3.801 \times 10^{-4} \) square meters.

Pressure Formula

Pressure measures how much force is applied over a certain area. You calculate it using the following formula:

• \( P = \frac{F}{A} \)

where \( P \) represents pressure, \( F \) is force, and \( A \) is the area. This formula shows that pressure increases if you apply more force or decrease the area.

In the syringe problem, you're calculating how much pressure the liquid feels when the nurse presses on the piston. By substituting the force (42 N) and the calculated area (\( 3.801 \times 10^{-4} \) m²) into the formula, you get the pressure increase, which is approximately 110,497.37 Pascals.

Pascal (Pa)

The Pascal (Pa) is the unit of pressure used in science. It tells us how much force is applied on one square meter of an area. One Pascal is equal to a force of one Newton acting on an area of one square meter.

This means that in situations like the syringe, where pressure needs to be calculated and understood, the result is often expressed in this unit. For our exercise, when the nurse applies force to the syringe, the pressure increase is calculated to be about 110,497.37 Pa.

Understanding and using the Pascal helps in accurately discussing and calculating pressures in various scientific and practical applications.