Geared wheel torque and energy conservation analysis

[LCSphysicist]

Homework Statement

I am trying to solve a problem by torques and conversation of energy, but the answers don't match.

Relevant Equations

t = rf
E = constant

By third Newton's law, i would imagine that fa = -fb, so
and assuming f to be the average value

torque in A
-ra*f*t = Waf - Wao
torque in b
rb*f*t = Wbf

and

Ia*wo² = Ib*wbf² + Ia*waf²

it Looks right to you? I really appreciate this method that i thought, but the answers don't match, and i don't know what is wrong.

THe justification to use the conservation of mechanic would be: The work done by one force will be equal to the work done by the reaction force, but opposite in signal, so it cancel.

 

[Lnewqban]

You could treat gear A as a flywheel.
There are many industrial presses and shear machines that use that principle for conforming and cutting metal.

Please, see:
https://www.engineeringtoolbox.com/flywheel-energy-d_945.html

The kinetic energy of gear A (Joules) will have to keep moving both gears after contact is established.

$E_{kA}=(1/2)(I_A)(\omega_0)^2$