question_answer

The magnitude of moment of a couple in certain direction is G and in a direction orthogonal to this direction (anti-clockwise) is H. The magnitude of moment of couple in anti-clockwise direction at an angle \[\theta \] to the critical direction is given by

A)  \[G\,\cos \,\theta -H\,\sin \theta \]

B)  \[-G\,\cos \,\theta +H\,\sin \theta \]

C)  \[G\,\cos \,\theta +H\,\sin \theta \]

D)  \[G\,sin\,\theta +H\,\cos \theta \]

Correct Answer: C

Solution :

Moment of couple in certain direction \[G=P.\,\,\,\,\,AM=Pd\,\,\sin \alpha \] [where P is force and d \[\sin \,\alpha \] is arm of couple] Moment of couple,  \[H=Pd\,\,\sin \,({{90}^{o}}+\alpha )\] \[=Pd\,\,\cos \,\,\alpha \] when force is turned through an angle \[\theta ,\] then moment of couple \[=Pd\,\,\sin \,(\alpha +\theta )\] \[=Pd(\sin \alpha \,\cos \theta \,+\cos \,\alpha \,\sin \theta )\] \[=\cos \,\theta \,(Pd\,\sin \alpha )+\sin \theta (Pd\,\cos \,\alpha )\] \[=G\,\,\cos \,\,\theta +H\,\,\sin \theta \]