# Pressure on each leg - why multiply the area?

Here's the problem (From here):

"A large man sits on a four-legged chair with his feet off the floor. The combined mass of the man and chair is 95.0 ππ. If the chair legs are circular and have a radius of 0.5 ππ at the bottom, what pressure does each leg exert on the floor?"

So, from the explanation, if we want to find the pressure exerted on "each" leg, we use

$$F=\frac{P}{4A}\tag{1}$$

Why is that the case? Why do we multiply the area by 4 when we want to find the pressure on "each" leg? My guess is that it's not multiplying the area by 4, but instead dividing the force by 4 since the force is equally distributed. I don't know if that's correct. Any (very throughout, if possible) explanation is much appreciated!

• Comment to the post (v3): Eq. (1) does not look right dimensionally. Oct 20, 2020 at 9:42
• "My guess is that it's not multiplying the area by 4, but instead dividing the force by 4 since the force is equally distributed. " And why do you think that both the explanations aren't equally valid? Oct 20, 2020 at 10:54

Yes your intuitions are right. You have a weight say $$W$$ supported by $$4$$ legs and thus each leg applies and experiences (from Newton's third law) a force of $$F = \frac{W}{4}$$ if the forces are uniformly distributed.

Also we know that

$$Pressure = \frac{force }{area}$$

The above equation means that you are uniformly distributing the force on a body among its all possible unit areas i.e if the total area is say $$5m^2$$ then the number of possible units of unit area is $$5$$ and thus you are now dividing the force among those $$5$$ units uniformly.

So to calculate pressure in your case , you need to first divide the force onto each leg and then divide each leg into more smaller units .

Its like dividing each leg of the chair into more smaller legs :) and then finding force on those smaller legs.

Hope it helps βΊοΈ.

Well, note that the dimension of this expression is not even valid (I assume that $$P$$ is the pressure and $$F$$ is the force here). So I think that you should not rely on this expression.

To solve the problem you can follow different routes:

• You can think as if the whole mass pressed against a surface of $$4A$$. In that case the pressure under the legs of the chair is just $$P = \frac{F}{4A}$$. Note, that the pressure is the same under every leg.
• You can assume, that the force is homogeniously distributed among the legs of the chair, thus each leg is pushed down by $$F_{leg} = \frac{F}{4}$$. Dividing this force by the surface of one leg you arrive at the same expression: $$P= \frac{F}{4A}$$.

Actually both of these reasonings is acceptable. It depends on your taste which one you choose.

• If I use the first route, does that mean I will get the pressure of the whole chair - for every leg - instead of each leg? If that's the case then why is the pressure exerted on the "chair" is equal to the pressure exerted on "each leg of the chair?" Oct 20, 2020 at 8:28
• I think you are thinking about pressure as it was an additive quantity. It is, but not in the same way as force is. You can add pressures if they are exerted on the same surface, but that comes from the addition of forces on that surface. If you can add pressure in the way you think about it, you could also just decompose the surface of one leg of the chair as a sum of $n$ number of $A_0$ surfaces. Adding up the pressures exerted on every surface, you would end up with a pressure as large as you want, based on $n$. Oct 20, 2020 at 20:19