Torque
Observing certain aspects of our everyday lives, we can feel that force alone is not enough to predict the changes in the state of motion of a rotating object. A longer spanner loosens a bolt more easily; the hinges and the doorknob are placed near opposite edges of the door. From these, we can conclude that the same force has greater effect if it acts at a greater distance from the axis. We call this distance the lever arm or moment arm of the force.
Line of action of a force
The line of action of a force is an imaginary line extending out both ends of the vector representing the force.
Lever arm | moment arm
The lever arm of a force is the perpendicular distance from the axis of rotation to the line of action of the force:
\[d = r \sin\vartheta,\]
where \(\mathbf{r}\) is the position vector from the fulcrum to the point of application of the force and \(\vartheta\) is the angle between vectors \(\mathbf{r}\) and \(\mathbf{F}.\)
The expression for the lever arm contains the angle between the position vector and the force in order to account for the fact that the same force acting at the same distance from the axis can have different effect if it acts in a different direction. If you exert a force at the doorknob in a direction perpendicular to the door (\(\vartheta = 90° \Rightarrow \sin\vartheta = 1\)), you can have maximum rotational effect, whilst exerting the same force parallel to the door (\(\vartheta = 0° \Rightarrow \sin\vartheta = 0\)) will not get the door open.
This rotational effect of a force is the product of the force and the lever arm. It is called torque. Substituting for the lever arm, we can see that the magnitude of the torque is
\[\tau = F \cdot d = r F \sin\vartheta.\]
Looking at this expression, we can see that it is identical in form to the vector product \(\mathbf{r} \times \mathbf{F}.\) Indeed, torque is a vector quantity, although it is slightly more difficult to interpret its direction.
Torque | moment of force
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Torque is the tendency of a force to rotate an object about an axis or fulcrum. It is defined as the vector product of the position vector \(\mathbf{r}\) measured from the fulcrum to the point of application of the force and the force vector \(\mathbf{F}\): \[\mathbf{τ} := \mathbf{r} \times \mathbf{F}.\] The torque vector is parallel to the axis of rotation and using the right-hand rule, we can tell the direction of the rotation as well: if the thumb of the right hand points to the direction of the torque, the fingers curl in the direction of the rotation. When more than one force acts on a rigid object, their torques add up as vectors and can be substituted by a single torque which is the vector sum of all torques. The SI unit of torque is the newton metre, which is not identical with the joule: \[\left[\mathbf{τ}\right] = 1\,\mathrm{N} \cdot \mathrm{m} \neq 1\,\mathrm{J}.\] |
Direction of torque in rotation about a fixed axis
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Though torque is a vector, for rotation about a fixed axis it is always parallel to the axis of rotation, so a \(\pm\) sign is enough to represent its vector nature. One possible convention is to take the torque with positive sign if it rotates anticlockwise and with negative sign if it rotates clockwise. |
