Understanding the relationship between pressure, force, and area is essential for every student of physics. Once you get the hang of it, this relationship can seem quite intuitive, and you’ll be able to use it to explain a lot of every-day occurrences. For example, it can tell us why its easier to pierce a loaf of bread with a sharp knife rather than a blunt one. Nevertheless, it’s important to have a clear understanding of this relationship, and not to fall into easy errors. In this post we’ll look at the equation that links pressure, force, and area in a bit of detail. To help you get really comfortable with it, we’ll then look at its application in some exam style questions.
The relationship between force, area, and pressure is given by the equation:
Pressure = Force / Area
Pressure is measured in Pascals (Pa), Force in Newtons (N), and Area in metres squared (m²).
To be more precise, we can express the above equation by saying that the pressure exerted on a surface is equal to the force applied to that surface divided by the area over which the force acts. We are only ever interested in the area over which a force is actually acting, the area of the whole surface experiencing the pressure isn’t relevant. So, for example, if a force is acting on one corner of a table, to work out the pressure on the table, we only need to take into account the area of that corner, not the area of the whole table.
The equation for pressure, which we’ve just looked at, can seem quite intuitive once you get a good grip on it. To show this, let’s look at some everyday examples which demonstrate this equation in action.
The equation can explain why it’s easier to cut bread with a sharp knife than with a blunt knife. The blade of a knife, however sharp, will always have a very small and relatively flat surface at its edge. A sharper knife, however, will have a much smaller flat surface than the blunt one. This means that, when you cut with the sharp knife, the area which comes in contact with the bread is lower than when you use the blunt one. If we use the equation above, we see that if you apply the same force in cutting, the pressure exerted by the sharp knife will be greater that by a blunt knife, due to the fact that a smaller area is coming in contact with the bread. This is why it’s easier to cut bread with a sharp knife.
You may have also heard that if you find yourself on thin ice, on a lake perhaps, its safest to lie down and crawl or roll to safety. When you lie down on the ice, the force you exert on it doesn’t change. The area over which that force is applied, however, will go from being the area of your shoes to the much larger area of your whole body. We learnt above that pressure is equal to the force acting divided by the area over which it acts. In this situation, the force is staying the same, but the area is increasing. This means that the pressure exerted on the ice will decrease as a result; meaning it is much less likely to break.
To give one last example, we can also increase the pressure exerted on a surface by simply increasing the force applied on it, without decreasing the area over which it acts. For example, if you push on a wall with both your hands, you can increase the pressure that you exert on that wall simply by pushing harder (i.e., by exerting more force).
Exam Style Question
Now, let’s look at an example style question to test what we’ve learnt:
'Imagine a cylindrical can, which is filled with baked beans, is resting on a table. The can and its contents have a mass of 0.6 kg. The bottom of the can has a radius of 0.02 m. Work out the pressure exerted by the can on the table.’
Here we’ll need to use our equation: Pressure = Force / Area.
First, we’ll work out the area over which the force is exerted. As the can is a cylinder, its bottom will be a circle. The equation for the area of a circle is: Area = π x radius².
We know the radius is 0.02 m, and we can treat π as roughly equal to 3.
This leaves us with: 3 x 0.02² m = 0.0012 m²
Now we need to work out the force. We know from elsewhere in the physics syllabus that the equation for force is force = mass x acceleration.
The mass is 0.6 kg. We can take the acceleration due to gravity to be roughly 10 m/s².
This leaves us with: 0.6 kg x 10 m/s² = 6 N
Now we’ve worked out the force and the area, we can use these values to work out the pressure.
6 N / 0.0012 m² = 5000 Pa
So, now we’ve worked out the pressured to be 5000 Pa.
[It should be noted that here we treated pi as 3 and acceleration due to gravity as 10 m/s² to make the maths easier, though you may have to use more significant figures for GCSE exams.]
In this post we’ve looked at the equation which relates pressure, force, and area. We’ve seen how this equation can give us intuitive explanations of everyday phenomena. Relating equations like these to your ordinary experience can help you to get a more thorough understanding of them. Hopefully now you’ll be able to put this equation to use in your own work.
Here we have only been able to study the pressure exerted by solid objects onto surfaces. During GCSEs, you'll mostly only have to use the equation for pressure which we’ve discussed to study these kinds of situations. As such, when you study pressure in liquids or gases, you will learn a different set of equations. Nevertheless, having a good grasp of the equation for pressure on surfaces will help you to understand pressure in liquids and gases, as the underlying physical processes in all these situations are similar.
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Logic Enthusiast is an independent writer and is studying for an MA in Philosophy at the University of Edinburgh. He is particularly interested in Logic and the Philosophy of Science.