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13.1: Torque

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  • Newton’s second law \(\vec{F}=m\vec{a}\) has a rotational analogue. When a force \(\vec{F}\) is exerted at a location \(\vec{r}\) measured from some axis of rotation (e.g., the bolt in Figure \(\PageIndex{1}\)), then the cross product \(\vec{r}\times\vec{F}\) is called the torque, \(\vec{T}\). The cross product is defined in Equation 2.1.1, and is derived in detail in section D.3.1. For now, it is a vector perpendicular to both \(\vec{F}\) and \(\vec{r}\), with direction given by the right-hand rule. The magnitude is


    where \(\phi\) is the angle between \(\vec{r}\) and \(\vec{F}\).

    Figure \(\PageIndex{1}\): Definition sketch for Newton’s second law in rotational form. A force \(F\) is exerted at a distance \(r\) from the axis of rotation, changing the angle \(\theta\).