A couple is of moment G and the force forming the couple is P.

Question

A couple is of moment $\vec G$ and the force forming the couple is $\vec P$. If $\vec P$ is turned through a right angle, the moment of the couple thus formed is $\vec H$. If instead, the forces $\vec P$ are turned through an angle $\alpha$, then the moment of the couple becomes:

Accepted Answer

With moment $=|\vec r||\vec p|\sin(	ext{angle})$: $G=|\vec r||\vec p|\sin	heta$, $H=|\vec r||\vec p|\sin(90^{\circ}+	heta)=|\vec r||\vec p|\cos	heta$.

Turning by $\alpha$ instead: new moment $=|\vec r||\vec p|\sin(	heta+\alpha) = |\vec r||\vec p|\sin	heta\cos\alpha+|\vec r||\vec p|\cos	heta\sin\alpha = G\cos\alpha+H\sin\alpha$

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