- #1
Aleoa
- 128
- 5
I'm trying to derive the lever law by myself, however, I'm stuck. Please follow the logic of my calculations.
Every object in the picture has the same mass. I want to prove that, under the effect of the gravitational force, I can replace the objects in A and C with the two objects in B, and obtain a system that behaves in the same way.
Since the same gravitational force acts in the objects, in an infinitesimal time interval [itex]dt[/itex] they all get the same [itex]dv[/itex]. However, the correspondent infinitesimal change in angular velocity also depends on the distance between the object(s) and the fulcrum.
If i prove that in an [itex]dt[/itex] the [itex]d\omega[/itex] due to the objects in A and C is equal to the [itex]d\omega[/itex] due to the two objects on B, I'm done.
[tex]d\omega_{A}+d\omega_{C}=d\omega_{B}[/tex]
that is
[tex]\frac{dv}{OA}+\frac{dv}{OA+D}=\frac{2dv}{OA+\frac{D}{2}}[/tex]
where O is the fulcrum and D is the distance between C and A.
However, doing some calculations i found that...
[tex]\frac{dv}{OA}+\frac{dv}{OA+D}\neq\frac{2dv}{OA+\frac{D}{2}}[/tex]
Maybe two single masses in A and C have not the same effect as a double mass in B, however according to Spivak:
Every object in the picture has the same mass. I want to prove that, under the effect of the gravitational force, I can replace the objects in A and C with the two objects in B, and obtain a system that behaves in the same way.
Since the same gravitational force acts in the objects, in an infinitesimal time interval [itex]dt[/itex] they all get the same [itex]dv[/itex]. However, the correspondent infinitesimal change in angular velocity also depends on the distance between the object(s) and the fulcrum.
If i prove that in an [itex]dt[/itex] the [itex]d\omega[/itex] due to the objects in A and C is equal to the [itex]d\omega[/itex] due to the two objects on B, I'm done.
[tex]d\omega_{A}+d\omega_{C}=d\omega_{B}[/tex]
that is
[tex]\frac{dv}{OA}+\frac{dv}{OA+D}=\frac{2dv}{OA+\frac{D}{2}}[/tex]
where O is the fulcrum and D is the distance between C and A.
However, doing some calculations i found that...
[tex]\frac{dv}{OA}+\frac{dv}{OA+D}\neq\frac{2dv}{OA+\frac{D}{2}}[/tex]
Maybe two single masses in A and C have not the same effect as a double mass in B, however according to Spivak: