[Physics] Why can you add moment vectors

forcesnewtonian-mechanicstorque

My book says the following:

If the body is acted upon by a system of forces, the resultant moment of the forces about point O can be determined by vector addition of the moment of each force. The resultant can be written symbolically

$$(M_R)_O = \Sigma (r \times F)$$

I don't understand why you can add the vectors. Shouldn't you just calculate the magnitude of each moment vector, if its counterclockwise make the magnitude positive, if its clockwise leave it positive and add the magnitudes?

For example if you have two moment vectors:

$$\vec{M_1} = (a,b,c)$$ (CLockwise)
$$\vec{M_2} = (c,d,e)$$ (Anticlockwise)

So you know that there is a moment of magnitude $| \vec{M_1} |$ acting clockwise and a moment of magnitude $|\vec {M_2} |$ acting counterclockwise. So why can't you just do the following:

$$ x = \pm | \vec{M_1} | \pm |\vec {M_2} |$$
$$\text{If } x > 0, \text {the moment is counterclockwise, if its negative its clockwise} $$

Where the sign of the moments depend on whether they are clockwise or anticlockwise.
Why is the above method invalid? The methods are obviously not the same since adding two vectors isn't equal to adding the magnitude unless the vectors are collinear.

Also, if you do add the vectors and get vector $M_3$, how do you know if $M_3$ is clockwise or counterclockwise?

Best Answer

So why can't you just do the following:

$$ x = \pm| \vec{M_1} | \pm |\vec {M_2} |$$ $$\text{If } x > 0, \text {the moment is counterclockwise, if its negative its clockwise} $$

You notice that $x$ will never get negative using this method but will only gets bigger and bigger, right? I think you misunderstand that more negative means smaller in magnitude, whereas it is actually bigger in magnitude, just in opposite direction.

Moments are vector quantities because the result may be smaller when they are summed up. They can be "cancelled" out.

Also, if you do add the vectors and get vector $M_3$, how do you know if $M_3$ is clockwise or counterclockwise?

It depends on your definition on the directions. If you say "OK, I'll define clockwise as positive value", then all your clockwise moments must be positive, while anticlockwise must be negative. When you add the vectors and get the result $M_3$, just look at its sign. If it is positive then you know it is clockwise; if it is negative then you know it is anti-clockwise.

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