Resultant Torque independent of origin?

[MrLiou168]

Homework Statement

Let a system of forces (F1,...Fn) act on a body at points (x1,...xn) respectively. Assume that the resultant or net force vanishes (sum of forces = 0)

Show that the resultant torque of this system is independent of the choice of origin, i.e. for 2 different origins x0 and x0', we have T = T' where:

T = [summation](xi-x0) X Fi and T' = [summation](xi-x0') X Fi

Homework Equations

T = F X d

The Attempt at a Solution

I have very little idea as to how to approach this problem, other than the T = Force X distance, and perhaps that the solution may have something to do with a bunch of couple moments that all equate to the same value, regardless of origin. Any help greatly appreciated!

 

[tiny-tim]

Hi MrLiou168!

T = [summation](xi-x0) X Fi and T' = [summation](xi-x0') X Fi

ok, now subtract one from the other:

you get … ? ​

 

[MrLiou168]

Hi MrLiou168!


ok, now subtract one from the other:

you get … ? ​

Thanks for the reply! But not quite, I'm a bit of a dim light most of the time... So assuming T = T', then T-T' = 0...

And subtracting the 2 equations nets me x0 - x0'. Therefore, is it sufficient to simply state that x0 - x0' = 0 to prove that the resultant torque is independent of choice of origin?

Thanks!

 

[tiny-tim]

… subtracting the 2 equations nets me x0 - x0'.

nooo …

what about all those forces? (∑ F_i) ​

 

[MrLiou168]

Oh... sorry I didn't realize the F's were summed as well...

So I have (x0 - x0') X Fi = 0? And since Fi is not zero, the x0 = x0'...?

 

[tiny-tim]

(try using the X₂ button just above the Reply box )

So I have (x0 - x0') X Fi = 0? And since Fi is not zero, the x0 = x0'...?

hmm, you're really confused …

you don't know T - T' = 0, that's what you're trying to prove!

all you have proved is T - T' = (x₀ - x₀') X (∑ F_i) …

now, what do you know about ∑ F_i ? ​

 

[MrLiou168]

A ha thank you for your patience! OK so one of the givens is that [sum] Fi = 0...

Therefore the right side of the equation nets zero and we have proven that T-T' = 0 and thus T = T' correct?

 

[tiny-tim]

correct!

 

[MrLiou168]

Tim, thank you very much - you were extremely helpful!