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Chapter 10: Problem 1

If you get jabbed by a pencil, why does it hurt so much more if it's with the sharp point rather than the eraser? Explain in terms of the concepts of force and pressure.

Short Answer

Expert verified
It hurts more when jabbed by the sharp point because pressure is force per unit area, and the sharp point has a much smaller area than the eraser, resulting in a higher pressure for the same amount of force.

Step by step solution

01

Understanding Force

Recognize that when you are jabbed by a pencil, a force is applied. This force is essentially the same whether it is applied by the sharp end or the eraser end of the pencil.
02

Defining Pressure

Understand that pressure is defined as force per unit area. Mathematically, it can be represented by the equation: \( P = \frac{F}{A} \), where \( P \) is pressure, \( F \) is force, and \( A \) is the area over which the force is distributed.
03

Comparing Contact Areas

Consider the differences between the contact areas of the sharp point and the eraser. The sharp point of a pencil has a much smaller contact area compared to the eraser.
04

Applying the Concept of Pressure to the Situation

Since the area of the sharp point is smaller, and the force is the same when jabbed with the same magnitude of force, the pressure exerted by the sharp point is far greater than that of the eraser. Higher pressure at the sharp point leads to a more intense sensation of pain.

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Key Concepts

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

Understanding Force
When you think of force, imagine a push or pull acting upon an object. This push or pull can cause the object to start moving, stop moving, or change its direction of motion. In the scenario of being jabbed by a pencil, the force is the amount of push that the pencil exerts on your skin. It can be gentle or intense, depending on how much effort is applied behind the jab. It’s important to note that the force itself does not change whether you are jabbed with the sharp end or the eraser — what changes is how this force is distributed, which leads to different levels of pain.
Pressure Definition
To grasp the concept of pressure, you can think of it as the force applied over a particular area. Formally, pressure is defined as the force exerted per unit of area. It's like how a bed of nails can support a person lying down without injury; the force of the person's weight is spread out over many points, resulting in lower pressure. However, if you were to step on a single nail, all the force of your weight would concentrate on a tiny area, increasing the pressure and likely causing pain. With pressure, it's not just the force that matters, but also how widely or narrowly it's spread out.
Contact Area
Contact area refers to the size of the surface over which a force is distributed. This is a critical factor when considering the effects of pressure. A larger contact area spreads out the force and results in a reduced pressure, akin to lying on a bed of nails as previously mentioned. A smaller contact area, on the other hand, means that the same amount of force is applied over a much smaller surface, greatly increasing the pressure. That's why being jabbed with a sharp pencil point—a very small contact area—hurts much more than being jabbed with the eraser end.
Pressure Calculation
To quantitatively determine pressure, we can use the mathematical equation
\[ P = \frac{F}{A} \],
where \( P \) is pressure, \( F \) is force, and \( A \) is the contact area. If you know the force and the area, you can calculate the pressure. For instance, if you are jabbed with a pencil, whether the force comes from the sharp point or the eraser, you can use this formula to find the exact pressure by substituting in the force exerted and the respective contact area. This straightforward calculation reveals why a sharp point causes more pain — it has to do with the higher pressure due to the smaller contact area.

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