# What is the torque formula

## Turning movements

If you turn a screw with a wrench, you need a force that is applied via the lever arm of the wrench handle. The further outside the wrench you attack, the lower the force must be to tighten the screw. Applying the force to the wrench as vertically as possible also reduces the effort required. The force \ (\ vec F \) has a direction and is therefore a vector. The distance of the force application point A from the axis of rotation D is also a vector \ (\ vec r \). If one wants to calculate the magnitude of the torque M without using vectors, one uses the distance \ (a \) of the pivot point from the line of action of the force and multiplies it by the amount of force \ (F \); it applies

\ [M = a \ cdot F \]

This distance \ (a \) can be calculated using the trigonometric relation \ (a = r \ cdot \ sin \ left (\ alpha \ right) \) from the radius vector \ (\ vec r \) (distance of the force application point to the pivot point) and calculate the angular width \ (\ alpha \) of the angle between the force vector \ (\ vec F \) and the radius vector \ (\ vec r \) without further use of the vector term, so that applies

\ [M = r \ cdot F \ cdot \ sin \ left (\ alpha \ right) \]

**Note:** In the figure on the right the angular width \ (\ alpha \) is larger than \ ({90 ^ \ circ} \). Therefore, the calculation of the route length \ (a \) here actually results in \ (a = r \ cdot \ sin \ left (180 ^ \ circ - \ alpha \ right) \). But since \ (\ sin \ left ({180 ^ \ circ - \ alpha} \ right) = \ sin \ left (\ alpha \ right) \) always applies, the above calculation method \ (a = r \ cdot \ sin \ left (\ alpha \ right) \) to the correct result.

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