What is the relationship between torque and net forces in a rotating system?

[Fibo112]

Homework Statement

A man of Mass M stands on a railroad car that is rounding an unbanked turn of radius R at speed v. His center of mass is height L above the car, and his feet are distance d apart. The man is facing the direction of motion. How much weight is on each of his feet.

Homework Equations

I have solved the problem too ensure that the torque about the center of mass is zero. What I don't understand is why the torque about the center of mass being zero ensures that the torque about any point is zero. I know that this is the case if there are no net forces acting, but that isn't the case here.

The Attempt at a Solution

 

[haruspex]

why the torque about the center of mass being zero ensures that the torque about any point is zero.

Suppose there is no net moment about some point: $\Sigma \vec {F_i}\times\vec {r_i}=0$.
Consider an axis at vectorial position $-\vec a$ from the first. The sum of moments about that point is $\Sigma \vec {F_i}\times(\vec {r_i}+\vec a)=\Sigma \vec {F_i}\times\vec {r_i}+\Sigma \vec {F_i}\times\vec a =(\Sigma \vec {F_i})\times\vec a$.
So if there is no net force then the sum of moments is the same everywhere.

 

[scottdave]

If there was a nonzero torque around a certain point, you would see the man spinning (accelerating) about that point.

 

[Fibo112]

Suppose there is no net moment about some point: $\Sigma \vec {F_i}\times\vec {r_i}=0$.
Consider an axis at vectorial position $-\vec a$ from the first. The sum of moments about that point is $\Sigma \vec {F_i}\times(\vec {r_i}+\vec a)=\Sigma \vec {F_i}\times\vec {r_i}+\Sigma \vec {F_i}\times\vec a =(\Sigma \vec {F_i})\times\vec a$.
So if there is no net force then the sum of moments is the same everywhere.

But there is a net force isn't there?

 

[haruspex]

But there is a net force isn't there?

Good point.
There is a net force through the man's mass centre, providing the centripetal acceleration.
That does not have any moment about the mass centre, so there is no net torque about that point. But there is a net torque about points not in that line of action.
Say the train is curving to the left. The net force acts to the left. Taking moments about a point between his feet, there is an anticlockwise torque.
Consider the centre of arc of the curve his feet are following. The man has angular momentum about that point. Because his mass centre is above the point, that angular momentum vector is not vertical. The torque produces precession, just like a gyroscope. The angular momentum vector changes direction as the man follows the curve.