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Chapter 12: Problem 107

A useful instrument for evaluating fibromyalgia and trigger-point tenderness is the doloriometer (or algorimeter). This device consists of a force meter attached to a circular probe that is pressed against the skin until pain is experienced. If the reading on the force meter is \(14.5 \mathrm{~N}\) ( \(3.25 \mathrm{lb}\) ) and the diameter of the circular probe is \(1.39 \mathrm{~cm}\), what is the pressure applied to the skin? Give your answer in pascals.

Short Answer

Expert verified
The pressure applied is approximately 95,492 Pa.

Step by step solution

01

Understand the problem

We are given a force of \(14.5\, \text{N}\) applied by a circular probe with a diameter of \(1.39\, \text{cm}\). Our task is to calculate the pressure in pascals.
02

Convert diameter to radius

The diameter of the circular probe is \(1.39\, \text{cm}\). First, convert this to meters to use standard units: \(1.39\, \text{cm} = 0.0139\, \text{m}\). The radius is half of the diameter: \(r = \frac{0.0139}{2} = 0.00695\, \text{m}\).
03

Calculate the area of the circular probe

Use the formula for the area of a circle, \(A = \pi r^2\), where \(r = 0.00695\, \text{m}\). Calculate the area:\[A = \pi (0.00695)^2 = \pi \times 4.83025 \times 10^{-5} \, \text{m}^2 \approx 1.518 \times 10^{-4} \, \text{m}^2\]
04

Calculate the pressure

Pressure is defined as force per unit area, \(P = \frac{F}{A}\). Use the force \(F = 14.5\, \text{N}\) and the area from Step 3:\[P = \frac{14.5}{1.518 \times 10^{-4}} \, \text{N/m}^2\]\[P \approx 95492 \, \text{Pa (Pascals)}\]
05

Present the final result

The pressure applied to the skin is \(95,492\, \text{Pa}\).

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Key Concepts

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

Understanding the Area of a Circle
When calculating the area of a circle, it is essential to use the formula: \(A = \pi r^2\). This formula expresses the area \(A\) in terms of the circle's radius \(r\). The radius is half the diameter, which is a critical detail to ensure accurate calculations.
For this exercise, the diameter of the circle given was \(1.39\, \text{cm}\), which needs to be converted to meters (\(0.0139\, \text{m}\)) for standard unit consistency.
Once you have the radius (\(0.00695\, \text{m}\)), plug it into the formula:
- Square the radius: \(0.00695^2\)
- Multiply by \(\pi\) (approximately 3.14159)
- Result in square meters for the area of the circle.
So, by following these steps, we calculated the area as approximately \(1.518 \times 10^{-4}\, \text{m}^2\). Understanding this formula is crucial not only for exercises like these but also for real-life applications where circular areas must be calculated.
Force Measurement Simplified
Force is a fundamental concept in physics and is measured in newtons (\(\text{N}\)). It represents the interaction that changes the motion of an object. In this exercise, the force was given as \(14.5\, \text{N}\) exerted by the doloriometer.
Understanding how force impacts pressure is critical:
  • Force is applied to a surface area to determine pressure.
  • The larger the force or the smaller the area, the greater the pressure.
Consider a situation where the same force is applied over a smaller or larger area:
  • A smaller area with the same force increases pressure, much like pressing down on a pencil versus your palm.
  • A larger area reduces pressure, distributing the force over a broader surface.
Grasping the concept of force measurement can help you appreciate how everyday tools and mechanical systems function, such as brakes on a bicycle or the application of pressure gauges.
The Importance of Unit Conversion
Unit conversion is vital in science and engineering fields. It ensures that all values are expressed in compatible units to avoid calculation errors. One common scenario requiring conversions is moving from one measurement system to another, for example:
  • from centimeters to meters,
  • or from newtons to pounds.
In this problem, the diameter measurement in centimeters had to be converted to meters to match the force units for pressure calculation:
- Convert units by dividing by 100 (since 1 meter = 100 cm).
Additionally, the force could have been given in pounds (as it sometimes is) and would require conversion to newtons (use 1 lb = 4.44822 N).
Properly converting units helps avoid misunderstandings and miscalculations when working between different unit systems. It is an essential skill in both academic environments and real-world scenarios.

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Most popular questions from this chapter

On the evening of July 23, 2001, a meteor streaked across the skies of Pennsylvania, creating a spectacular fireball before exploding in the atmosphere with an energy release equivalent to 3 kilotons of TNT. The pressure wave from the airburst caused an increase in pressure of \(0.50 \mathrm{kPa}\), enough to shatter some windows. Find the force that this "overpressure" would exert on a window whose dimensions are \(86 \mathrm{~cm} \times 115 \mathrm{~cm}\). Give your answer in newtons and in pounds.

A person floats in a boat in a small backyard swimming pool. Inside the boat with the person are several blocks of wood. Suppose the person throws the blocks of wood into the pool, where they float. (a) Does the boat float higher, lower, or at the same level relative to the water? (b) Does the water level in the pool increase, decrease, or stay the same?

A fluid flows through a pipe with a speed of \(1.8 \mathrm{~m} / \mathrm{s}\). The diameter of the pipe is \(2.7 \mathrm{~cm}\). Further along, the diameter of the pipe changes, and the fluid flowing in this section has a speed of \(3.1 \mathrm{~m} / \mathrm{s}\). What is the new diameter of the pipe?

At the beginning of an airline flight, you are instructed about the proper use of the oxygen masks that will fall from the ceiling if the cabin pressure suddenly drops. You are advised that the oxygen masks will work properly, even if the bags do not fully inflate. In fact, the bags expand to their fullest if cabin pressure is lost at high altitude but expand only partially if the plane is at low altitude. Explain.

Two identical springs are connected end to end. Is the spring constant of the resulting compound spring greater than, less than, or equal to that of each single spring? Explain.
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