Law of the lever: Conservation of energy or angular momentum

[greypilgrim]

Hi,

Some "derivations" of the law of the lever argue with conservation of energy: If one arm of the lever of length ##r_1## is pulled by a distance ##s_1## with force ##F_1##, the other arm moves by a distance ##s_2=s_1 \frac{r_2}{r_1}##. From conservation of energy ##E=F_1 s_1=F_2 s_2## it follows $$F_2=F_1 \frac{s_1}{s_2}=F_1 \frac{r_1}{r_2}\enspace.$$
However, the law of the lever also holds in static situations where ##s_1=s_2=0## and no work is being done and above derivation breaks down. A derivation that both includes moving and static situations uses the fact that all torques must vectorially add up to zero which follows from conservation of angular momentum.

So I wonder if the derivation using conservation of energy only works coincidentally, because energy and torque share the same unit. From a Noetherian perspective, the derivations are very different, the first following from homogeneity in time, the other from isotropy in space.

As a more general question, is it mere coincidence that energy and torque have the same unit or is there more to it?

 

[mfb]

You can consider virtual displacements if you like.
The limit for ##s_2 \to 0## is well-defined and gives the same result. The attempt to divide by zero is a purely mathematical problem.

 

[A.T.]

So I wonder if the derivation using conservation of energy only works coincidentally, because energy and torque share the same unit.

You can derive the static lever law without invoking the concept of torque, using only linear forces on a truss structure. There were several threads on this here.

 

[greypilgrim]

That's interesting, so the law of the lever can actually be derived either from conservation of energy, conservation of linear momentum OR conservation of angular momentum independently, hence by Noether's theorem either from homogeneity in time, in space or isotropy in space?