I am having a difficult time understanding when to use cos or sin when finding the moment of a force about a point. I understand that moment is $\rm force\cdot distance$. I have attached a picture of a arm with length $L$ where we need to find the moment due to the force of gravity. The answer is $-\frac{mgL}{2}\sin\theta$ and I have no idea why it is $\sin\theta$ and not $\cos\theta$ and what method to use for different problems to know if it is $\cos\theta$ or $\sin\theta$. Thank you![enter image description here]1


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    $\begingroup$ What is theta?... $\endgroup$ – DJohnM Nov 16 '17 at 6:47

Firstly, you need to specify where angle $\theta$ is defined.

From the answer, that it is $\sin\theta$, we can conclude that $\theta$ is the angle between $L$ and $g$. If it was defined as the angle between $L$ and horizontal axis, it would be $\cos\theta$ instead of $\sin\theta$.

A quick trick to do in such situations is to consider the extreme limits. You know that the angular dependence is sinusoidal, and you know that moment is zero if force and distance vectors are parallel. You also know that $\cos 0=1$ and $\sin 0=0$. That means, if the angle you use, say $\theta$, becomes zero when the vectors are parallel, then the angular dependence should be $\sin\theta$. If $\theta$ is $\pi/2$ when the vectors are parallel, then you need $\cos\theta$.


Put your finger along the vector( A ) and sweep it parallel to the line( L ) over which you need to get its component. The component of A along L will be cos of the angle swept by your finger( θ ). The component of A perpendicular to L will be Cos(90-θ) by same trick which is same as Sin(θ). Along with this, consider the frame of reference, the sign conventions, etc and you will never have to confuse with when to use Sin (θ) or Cos(θ).

Hope it helps.


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