# How fundamental is the Law of the Lever?

1. Oct 4, 2012

### holtto

Everyone knows about the law of the lever, in order for a see-saw to balance the torques must cancel each other.

The question is, how fundamental is it? Can the Law of the Lever be derived from Newton's three laws or is it a fundamental law in its own regard?

Some may say it stems from consv. of energy, but it still holds when no work is being done.

2. Oct 4, 2012

### Staff: Mentor

historically its been known far longer than Newtons laws and I would say that makes it more fundamental.

http://en.wikipedia.org/wiki/Lever

Can you derive it from Newtons laws?

Yes you can but Archimedes proved it using geometric methods long before Newton.

Your question is kind of a circular reasoning type question where we take one thing and derive another and vice versa meaning its meaningless to say which is more fundamental.

3. Oct 7, 2012

### holtto

what's the derivation from newton's laws? Teach me, shrfu.

4. Oct 7, 2012

### runningninja

I would think a very simplistic, calculus and vector free derivation would be to begin with
F=ma
Then multiply both sides by radius r.
Fr=mar
The left hand side simplifies to become torque. You can also restate r as r2/r. Thus:
τ=ma(r2/r)
I'll rewrite the right hand side of the equation.
τ=m(r2)*a/r
m(r2)=rotational inertia, or I. a/r= angular acceleration, or α. Thus:
τ=Iα.
Clearly, if linear acceleration is 0, angular acceleration must also be 0. This is the case if there is a constant linear velocity, as is the case with statics. A seesaw has a constant angular velocity, meaning that the angular acceleration must be zero, which means that the net torque must equal zero, which is pretty much the law of the lever.
This is not mathematically exhaustive because I used basic algebra to prove it rather than vector analysis.

5. Oct 8, 2012

### Staff: Mentor

Here's a website that discusses it further. Basically the law of the lever inspired the more general concepts of angular momentum and torque so that nowadays we can use CM theory to solve this and other cases as well.

Another example would be how the ancient Greeks determined the surface and volume formulas for a cone and a sphere. How did they do it?

The best answer I found is that they applied limit concepts to the problem. Nowadays we use calculus to determine these formulas which is also based on limit concepts now developed into calculus rules of integration and differentiation.

The ancient engineers also determined very accurate torsion formulas for catapults with twsted rope design that was based on the cube root of 100. How they determined the formula and tools is a mystery.

http://www.mlahanas.de/Greeks/war/Catapults.htm

So its like anecdotal and empirical ideas often more general theory that can be later used to derive the original results.

6. Oct 8, 2012

7. Oct 8, 2012

### Staff: Mentor

With the same argument, the concept that everything consists of earth, fire, water and air is more fundamental than the modern concept of elements?

Those geometric derivations use Newton's laws in an implicit way.

I would consider Newton's laws as fundamental, and the laws concerning lewers can be derived from them - the other direction is not possible.

8. Oct 8, 2012