I am confused about a problem in my textbook (Serway's College Physics, quick quiz 12.2 on the 9th edition). The problem considers an extended object that is acted on by three separate forces, all of which pass through the same point $P$:
It is apparent that the net force is not zero so this object is not in force equilibrium. However, the answer to this quick quiz claims that the object is in torque equilibrium. I don't quite agree with the answer.
It is true that if we choose the reference axis to pass through point $P$ the net torque is zero. Nevertheless, if we choose other reference axes, the net torque will no longer be zero. It is also claimed in the book that the net external torque on the object about "any" axis must be zero in order to achieve static equilibrium.
Let me do the math. There are three forces and let me call them $\mathbf F_1$, $\mathbf F_2$, and $\mathbf F_3$. First let us choose point $P$ to be the reference point. The net torque is $$\mathbf r_1 \times \mathbf F_1 + \mathbf r_2 \times \mathbf F_2 + \mathbf r_3 \times \mathbf F_3= 0+0+0=0,$$ where $\mathbf r_1$, $\mathbf r_2$, and $\mathbf r_3$ are the position vectors from $P$ to the application points of $\mathbf F_1$, $\mathbf F_2$, and $\mathbf F_3$, respectively.
If I instead choose another origin located R away from point P, the net torque is $$(\mathbf r_1+\mathbf R) \times \mathbf F_1 + (\mathbf r_2+\mathbf R) \times \mathbf F_2 + (\mathbf r_3+\mathbf R) \times \mathbf F_3 = \mathbf R \times (\mathbf F_1+\mathbf F_2+\mathbf F_3)$$ is apparently not zero.