This is an amalgamation of a few posts I've made in various threads to explain a bit about sharpness in relation to razors and knives, because I think it's quite an important thing to understand when using, and especially honing different kinds of blade.
Sharpness is not an absolute characteristic of an edge itself in which a single test can tell you whether one thing is 'sharper' than another, because it also depends on what you're cutting. If you take a razor and a kitchen knife and try to shave with them your razor will cut the hair very easily and closely, but not your skin. Whereas the knife may cut your skin, but probably not your hair very well. And neither are going fell a tree.
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Here's a rubbish drawing of a knife cutting a tomato. Where 'b' is the angle of your edge (sharpening angle), 'a' is 90 degrees minus 'b', and 'c' is 90 degrees minus the angle you're cutting the tomato at:
In order to make the knife cut the tomato well we need to increase the frictional force between the two. ***Everything about how anything cuts anything is about how friction works.***
F = uN.
The Frictional Force (F) equals the Coefficient of Friction (u) multiplied by the Normal Force (N). So to make the knife cut the tomato we want to try to increase either u or N, or both. Let's first look at N...
The main reason that ripe tomatoes are used often when testing the edge of a kitchen knife isn't because they're unusually hard or tough - the skin is fairly comparable to many other fruit and veg, it's because the flesh is unusually soft. And that massively decreases the Normal Force which is exerted upwards by the tomato against your knife. Here are another two rubbish drawings:
In the first the knife cuts straight down and the Normal Force acts straight up against it. But unfortunately you don't always want to neatly bisect a tomato; you cut it more like the second drawing, in which 'd' is the angle 'a' minus 'c' from the first pic. Meaning we only get a small proportion of the Normal Force, which I'm calling N1. We can see if we decrease the sharpening angle 'b', then we increase 'a', hence increase 'd', and subsequently increase the value of N1. So a more acute edge will cut the tomato better.
But there's a slight problem with that, because after you cut through your tomato you hit your chopping board, which is hard. And if your edge is too acute then it will be frail, and crumple when it hits the board, and now your knife is blunt. So we can't just take the edge angle down as low as possible to increase N to the max, we need to try to increase the Coefficient of Friction (u) in some way too...
The other reason tomatoes are used in this test is because the skin is quite smooth, and that decreases u. Try cutting a slippery wet tomato vs a dry one and you'll see the importance of the friction coefficient. But apart from making sure your tomato is dry there is another way to increase u - by finishing the knife on a lower grit stone to leave more 'teeth' on the edge, making it like a microscopic saw. And using a push or a pull cut (literally a sawing motion) rather than chopping straight down to amplify the effect.
So in fact: While we might think that finishing an edge at a higher grit will make it sharper, for many applications it doesn't.
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Now let's go back to looking at our razor...
A general purpose kitchen knife is designed to cut most foods well, whereas a razor is a highly specialised instrument designed to cut some things very well and other things very badly. You don't actually want a razor to be particularly good at cutting a tomato, because your face likewise has a relatively tough outer skin and soft flesh, and if you had a high coefficient of friction you'd cut yourself. It should cut a tomato better at bevel set than it does at finish.
So you refine and polish the edge with a series of progressively higher grit stones and you cover face in lather, which reduces 'u'. And you shave with light pressure at a low angle, which reduces 'N'. And all this means that your razor (hopefully) glides smoothly over your skin, while effortlessly cutting your hair.
---
If you take two different types of edged instrument that have apexes that meet - you can't really say that one is 'sharper' than the other. Knowing what you're trying to achieve is essential, because there are different kinds of sharp.
Sharpness is not an absolute characteristic of an edge itself in which a single test can tell you whether one thing is 'sharper' than another, because it also depends on what you're cutting. If you take a razor and a kitchen knife and try to shave with them your razor will cut the hair very easily and closely, but not your skin. Whereas the knife may cut your skin, but probably not your hair very well. And neither are going fell a tree.
---
Here's a rubbish drawing of a knife cutting a tomato. Where 'b' is the angle of your edge (sharpening angle), 'a' is 90 degrees minus 'b', and 'c' is 90 degrees minus the angle you're cutting the tomato at:
In order to make the knife cut the tomato well we need to increase the frictional force between the two. ***Everything about how anything cuts anything is about how friction works.***
F = uN.
The Frictional Force (F) equals the Coefficient of Friction (u) multiplied by the Normal Force (N). So to make the knife cut the tomato we want to try to increase either u or N, or both. Let's first look at N...
The main reason that ripe tomatoes are used often when testing the edge of a kitchen knife isn't because they're unusually hard or tough - the skin is fairly comparable to many other fruit and veg, it's because the flesh is unusually soft. And that massively decreases the Normal Force which is exerted upwards by the tomato against your knife. Here are another two rubbish drawings:
In the first the knife cuts straight down and the Normal Force acts straight up against it. But unfortunately you don't always want to neatly bisect a tomato; you cut it more like the second drawing, in which 'd' is the angle 'a' minus 'c' from the first pic. Meaning we only get a small proportion of the Normal Force, which I'm calling N1. We can see if we decrease the sharpening angle 'b', then we increase 'a', hence increase 'd', and subsequently increase the value of N1. So a more acute edge will cut the tomato better.
But there's a slight problem with that, because after you cut through your tomato you hit your chopping board, which is hard. And if your edge is too acute then it will be frail, and crumple when it hits the board, and now your knife is blunt. So we can't just take the edge angle down as low as possible to increase N to the max, we need to try to increase the Coefficient of Friction (u) in some way too...
The other reason tomatoes are used in this test is because the skin is quite smooth, and that decreases u. Try cutting a slippery wet tomato vs a dry one and you'll see the importance of the friction coefficient. But apart from making sure your tomato is dry there is another way to increase u - by finishing the knife on a lower grit stone to leave more 'teeth' on the edge, making it like a microscopic saw. And using a push or a pull cut (literally a sawing motion) rather than chopping straight down to amplify the effect.
So in fact: While we might think that finishing an edge at a higher grit will make it sharper, for many applications it doesn't.
---
Now let's go back to looking at our razor...
A general purpose kitchen knife is designed to cut most foods well, whereas a razor is a highly specialised instrument designed to cut some things very well and other things very badly. You don't actually want a razor to be particularly good at cutting a tomato, because your face likewise has a relatively tough outer skin and soft flesh, and if you had a high coefficient of friction you'd cut yourself. It should cut a tomato better at bevel set than it does at finish.
So you refine and polish the edge with a series of progressively higher grit stones and you cover face in lather, which reduces 'u'. And you shave with light pressure at a low angle, which reduces 'N'. And all this means that your razor (hopefully) glides smoothly over your skin, while effortlessly cutting your hair.
---
If you take two different types of edged instrument that have apexes that meet - you can't really say that one is 'sharper' than the other. Knowing what you're trying to achieve is essential, because there are different kinds of sharp.