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How is Torque Defined?

Answer #1

Level 3 (with higher mathematics)
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The torque \( \class{green}{\boldsymbol{M}} \) is defined as follows:

Definition of the torque
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The torque is the cross product of lever-arm length \( \boldsymbol{r} \) and force \( \boldsymbol{F} \).

Because of the cross product the torque vector is always perpendicular to \( \boldsymbol{r} \) and \( \boldsymbol{F} \)! Its magnitude \( |\boldsymbol{r}\times \boldsymbol{F}| = \class{green}{M}\) is the area spanned by the vectors \( \boldsymbol{r} \) and \( \boldsymbol{F} \).

Cross Product of Two Vectors
Cross product of any two vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\).

The magnitude of the torque, according to the definition of the cross product, is given by the following formula:

Magnitude of the torque
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Here \(\varphi\) is the angle enclosed by the vectors \( \boldsymbol{r} \) and \(\boldsymbol{F}\). The torque becomes maximum when the two vectors are orthogonal to each other. So if the angle is \(\varphi = 90^{\circ}\). Formula 2 then simplifies to the product of the force and lever-arm length.

Maximum torque
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