# Varignon's theorem (Principle of moments) failure?

"If many coplanar forces are acting on a body, then the algebraic sum of torques of all the forces about a point in the plane of the forces is equal to the torque of their resultant about the same point."

Consider the following case: Let's consider net moment about the point O.

Algebraic sum of torques gives us √2F*√2a + F*a = 3Fa

But resultant of the two forces = F in downward direction gives us the torque Fa

Clearly Varignon's theorem breaks down in this case. I've read elsewhere that Varignon's thorem doesn't apply for cases where a couple of forces is involved. And that's obvious. Because resultant of couple is 0 force which gives 0 torque - and that is wrong.

But in this case, the two forces don't seem to form a couple. The horizontal component of √2F force does form a couple with the bottom force, but that way, in a generalised problem with n forces, many components can form a couple. So how do we know when not to apply varignon's theorem?

• The forces have to be concurrent (see e.g. Varignon's theorem (mechanics) ). They are not concurrent in your case, so the theorem does not apply. Jul 17 '18 at 16:53
• @NickD concurrency is a necessary condition. However, I have paraphrased an alternate statement of the theorem from the same wiki link, and this doesn't talk about concurrency. Can you please confirm wiki is wrong? Jul 18 '18 at 9:18
• I don't know which wiki you quoted (I don't see a link) but the "coplanar" part above should probably read "concurrent". Also your statement about the couple is wrong: the resultant force may be 0 but the torque is not. Jul 18 '18 at 12:52
• @NickD I'm quoting the wiki link you added in your first comment. You're misreading the 0 torque statement. I'll edit the phrasing so it's clearer Jul 18 '18 at 14:25
• I checked the reference: the "coplanar" part is a mistake (which I fixed). Jul 18 '18 at 16:47