Why Is the Sum of Torques Always Zero Regardless of the Axis Location?

[something_about]

hi

If net force on the body is zero and also sum of torques is zero,then object doesn't rotate and wherever we put an axis, the sum of torques is always zero.

I know it's true and it does make sense, but can you show me mathematical proof (or vector proof ) that no matter where the axis is, the sum of torques is always zero?

In the following example no matter where we put an axis the sum of torques is zero(I already solve it, so no need for help there)

     _____|C|__________                                
     |A|           |B|

Board( weight of the board is negligible ) length is 4 meters and lays on top of boxes A and B. Third box C is on top of board and has mass 16 kilos. Length from A to C is 0.5 m.

thank you very much

 

[Astronuc]

For the system to be static, i.e. $\Sigma$F = 0 and $\Sigma$M = 0 (sum of moments).

The sum of forces is straight forward = F_A+F_B-F_C = 0 => F_A + F_B = F_C

OK then one does $\Sigma$M = 0

Take the moment about A and B separately.

At A, M_A = 0, because the moment arm of F_A is zero, and I am assuming no net torque applies, and one has M_C + M_B = 0. Take the + moment to be counter-clockwise, -l_C*F_C + l_B*F_B = 0.

The same method can be applied at B.

One needs one equation for each unknown otherwise the system is indeterminate.

 

[something_about]

but how does that prove that no matter where the axis is the sum of torques is zero?

Subquestion if I may...when we say the axis can be anywhere and sum is still zero...do we mean laying anywhere inside the body or can axis also lay outside the body?