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Illustration 11.1: Cross Product
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We talk about the magnitude of the torque as the amount of force perpendicular to the radius arm on which it acts. No radius arm, no torque. Torque is positive (out of the page) if F acts to rotate the object counterclockwise via the right-hand rule (RHR) and negative if F acts to rotate the object clockwise (again, via the RHR). Restart.
In order to mathematically describe torque, we must use the mathematical construction of the vector or cross product. Torque is the vector product of the radius vector and the force vector, r × F. The magnitude of the torque is r F sin(θ), and the direction of the torque is determined by the RHR. θ is the angle between the two vectors, and A and B are the magnitudes of the vectors r and F, respectively. Drag the tip of either arrow (position is given in meters). The red arrow is r and the green arrow is F. The magnitude of each arrow is calculated as well as the cross product.
The direction of the torque, r × F, is determined by the RHR (point your fingers toward r, curl them into the direction of F, and the direction that your thumb points is the direction of the torque. Therefore,
τ = r × F = r F sin(θ) with the direction prescribed by the RHR,
where r is the moment arm on which the force acts and F is the force.