# Are all machines linearly scalable?

For all machines (cars, elevators, computers, etc), when size, power requirements, dimensions are scaled by a constant N, will it work just as is?

Will a car with all its parts 10x larger still work like a normal car, just larger?

• For a very simple case, consider a body that generates heat relative to its volume, but can only transfer heat relative to its surface area. As you scale it up, it will get hotter and hotter until it breaks down. – raptortech97 Oct 7 '14 at 1:15
• Same question but for animals: physics.stackexchange.com/questions/72641/… – user10851 Oct 7 '14 at 6:01
• A car that is 10 times bigger doesn't 'work' because you can't reach the pedals. An elevator that is 10 times bigger wouldn't 'work' because it would occupy all stories of the building at once. A computer with a transistor that is 10 times bigger is probably comletely useless because the electricity has to travel through a very long path. – Dennis Jaheruddin Oct 7 '14 at 10:16
• even worse, a car that's 10 times as large in size has a volume of material that's 1000 times as large while not having 1000 times as much engine power... And the same with most things when scaled up. – jwenting Oct 7 '14 at 14:29
• Perhaps a better question would be "are there any machines which are linearly scalable which remain useful"? Consider the lever - a ten-foot-long lever might be very useful for moving something. A 100-foot-long lever? Maybe not so much... – Bob Jarvis Oct 7 '14 at 22:48

This is a great question. An influential early discussion of it was given in a 1959 talk by Richard Feynman, There's Plenty of Room at the Bottom. Basically the answer is no, machines are not linearly scalable. For example, lubrication doesn't work for very small machines. A general way of looking at this is that we have various physical quantities, and they scale in different ways. For example, area is proportional to length squared, while volume goes like length cubed. Because different things scale differently, what works on one scale doesn't work on another. A good example is the animal world -- after all, animals are a type of machine. A spider the size of an elephant would collapse under its own weight. This is because the strength of the animal's limbs go like the cross-sectional area, while the weight the limbs have to support is proportional to volume.

• tl,dr: Weight scales like $\text{length}^3$ but strength of an e.g. bone scales like cross sectional area $=\text{length}^2$. Therefore, as you make an animal or machine bigger, the strength to weight ratio goes down. – DanielSank Oct 7 '14 at 1:12
• An even earlier influential discussion was Discourses and Mathematical Demonstrations Relating to Two New Sciences (Galileo, 1638). As I understand it from second-hand accounts, one of the two sciences was the relation between material strength and scaling, and the other was kinematics. – Nathaniel Oct 7 '14 at 2:52
• The flip side of scaling a spider to the size of an elephant is scaling a shark to the size of a rotifer (< 1 millimeter). The shark would be easy prey for the rotifer because the miniaturized shark can't move. The thunniform locomotion used by sharks doesn't work at exceedingly low Reynolds numbers. – David Hammen Oct 7 '14 at 12:55
• Things get even more (or less, depending on your perspective) interesting when you scale down near Planck scale or up to a scale where relativity kicks in and propagation of pure mechanical motion from one part of the machine to another would take place spread out over years. – R.. Oct 7 '14 at 15:23
• Nice answer. I'll just add that for further reading the principle is often termed "allometry", "allometric scaling", or the "square-cube law". The canonical examples are often the bone structures and metabolic rates of a mouse and an elephant. – DeveloperInDevelopment Oct 7 '14 at 19:58