Consider, say, a rigid 2-D square sheet, of uniform mass density, initially occupying the square:
$$-2 \leq x_1, \ \ x_2 \leq 2$$ in the $x_1 x_2$-plane. The plane is a frictionless table, on which the sheet lies.
For simplicity, consider applying a pair of forces $\pm \vec{F}$, with $\vec{F} = (1,0)$, at: (1) the points $(-1, -1)$ and $(-1, -3/4)$, and alternatively: (2) the points $(1,1)$ and $(1, \frac{3}{4})$. The couples (1) and (2) have the same couple moment, so, by Varignon's theorem, should have the same rotational effect on the square sheet. So, in each case, (1) and (2), if the pair is maintained fixed in the frame of the sheet, the resulting rotary motion of the sheet should be the same. But this seems counterintuitive: wouldn't the sheet rotate about different centers with (1) vs. (2)? Am I tacitly assuming the presence of other forces (e.g., constraints introduced if we tried rotating the sheet by using a two-pronged fork)?
