# Scaling: Gravity and Friction

I understand how doubling the length of a shape quadrupes it's area and the analog in 3 dimensions. My question however relates to other physical quantities, for example gravitational field strength. Lets consisder some system under constant gravity. How will the be behaviour of the system change if we change the strength of the gravity. Is their a general way to tell? How does friction scale if we change the size? Are any of these questions related to tr units of these quantities? Eg. Acceleration due to gravity $\frac{m}{s^2}$ does this mean its somehow proportional to some distance? Is there a conceptual to think about this? The only scaling law articles I found related to length area and volume

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Gravity (just like the other fundamental forces-- ignoring GR's geometric interpretation of gravity for now) has a force law which depends on distance: $F\propto \frac{1}{r^2}$. – Danu Nov 12 '13 at 18:36
Yes, but the original question assumes constant gravity. So we are only considering a change in $g$, not in distance. – legrojan Nov 12 '13 at 18:53

There isn't a simple answer to your question. The scaling will be different in different situations. Let's take your example of gravity. The acceleration is given by:

$$a = G \frac{M}{r^2}$$

so $a$ scales as mass$^1$ and distance$^{-2}$. But consider some other quantity like the orbital period, which is given by:

$$T = 2\pi \sqrt{\frac{r^3}{GM}}$$

then $T$ scales as mass$^{-1/2}$ and distance$^{3/2}$. The only way to work out what the scaling will be is to sit down with a piece of paper and write down the relevant equations for the property you are interested in.

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How would we do it if we can't find a formula? I have been considering a small block doing a loop-the-loop with friction, and have managed to solve it, by approximation a differential equation. I'm looking to find an argument to show it is proportional to the radius of the loop. link – Michal Nov 12 '13 at 19:16
@Michal: You could try dimensional analysis. If that doesn't work and you can't find an EOM then you're stuck with experiment. – John Rennie Nov 12 '13 at 19:27
Sorry if it's a simple question but whats EOM? – Michal Nov 12 '13 at 19:30
@Michal: oops, sorry - Equation Of Motion – John Rennie Nov 12 '13 at 19:35
Okay, but umm... Isn't that mass$^{-1/2}$ at the last? :) – Waffle's Crazy Peanut Nov 14 '13 at 7:00

First of all, length, area and volume are related to each other through different powers of the same basic dimension (length, $L$): length is $L^1$, area is $L^2$, and volume, $L^3$. So if you change the fundamental length, obviously the other two scale accordingly, but we are talking of three different units here.

On the other hand, changing the magnitude of the gravity does not change the type of units of the problem. In fact, no problem has physical meaning if it changes under a different choice of units. When you increase the gravity, you simply make the gravitational force felt by the objects larger.

In the particular case of a frictional force: the formula is (to a first approximation) $$F_{fric}=k m g,$$ being $m$ the mass of the object, $g$ the gravity and $k$ is the frictional coefficient, which depends on both the object and the surface on which it moves, and is a number between 0 and 1. So if you multiply $g$ by a factor of 2, the problem will scale accordingly: the weight of the object will be twice as much, and the frictional force as well. But the physics behind the problem does not change at all.

If you were to change the object itself, and thus its mass $m$, the friction might change if you alter the shape or composition of the object. But if we consider a square block on a flat surface, making it bigger will not change the fundamental friction coefficient $k$, so the force will just be proportional to the changed mass.

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Sorry if I wasn't being clear, I wasn't asking for a specific case of a block on a surface, just a general to explain it conceptually. By increase in gravity I mean moving from planet to planet, how will it affect a complicated experiment - is there a general way to tell. – Michal Nov 12 '13 at 18:54
Yes, that is my point as well. You can take a ball on an inclined plane to the Moon, and the friction will be the same as here on Earth. Not the force amount, since $g$ is different, but the friction coefficient will not change. I talked about a block on a surface because it's a simple example: if you make it bigger, the contact between the object and a surface on which it moves will not change conceptually, it will just be larger. But the principle is general for any planet, any object, any surface. A universal law is just that: universal. – legrojan Nov 12 '13 at 19:08
OK, I think I understand now that you wanted a more general answer, not just a friction/gravity case. In that case, like John Rennie says, you need to look at the units of the particular problem. – legrojan Nov 12 '13 at 19:17