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Physics LibreTexts

18.5: Appendix 18A The Torques About any Two Points are Equal for a Body in Static Equilibrium

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When the net force on a body is zero, the torques about any two points are equal. To show this, consider any two points A and B . Choose a coordinate system with origin O and denote the constant vector from A to B by rA,B. Suppose a force Fi is acting at the point rO,i. The vector from the point A to the point where the force acts is denoted by rA,i, and the vectors from the point B to the point where the force acts is denoted by rB,i.

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Figure 18A.1 Location of body i with respect to the points A and B .

In Figure 18A.1, the position vectors satisfy

rA,i=rA,B+rB,i

The sum of the torques about the point A is given by

τA=i=Ni=1rA,i×Fi

The sum of the torques about the point B is given by

τB=i=Ni=1rB,i×Fi

We can now substitute Equation (18.A.1) into Equation (18.A.2) and find that

τA=i=Ni=1rA,i×Fi=i=Ni=1(rA,B+rB,i)×Fi=i=Ni=1rA,B×Fi+i=Ni=1rB,i×Fi

In the next-to-last term in Equation (18.A.4), the vector rA,B is constant and so may be taken outside the summation,

i=Ni=1rA,B×Fi=rA,B×i=Ni=1Fi

We are assuming that there is no net force on the body, and so the sum of the forces on the body is zero,

i=Ni=1Fi=0

Therefore the torque about point A , Equation (18.A.2), becomes

τA=i=Ni=1rB,i×Fi=τB

For static equilibrium problems, the result of Equation (18.A.7) tells us that it does not matter which point we use to determine torques. In fact, note that the position of the chosen origin did not affect the result at all. Choosing the point about which to calculate torques (variously called “ A ”, “ B ”, “ S ” or sometimes “O ”) so that unknown forces do not exert torques about that point may often greatly simplify calculations.


This page titled 18.5: Appendix 18A The Torques About any Two Points are Equal for a Body in Static Equilibrium is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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