12.1 Conditions for static equilibrium  (Page 2/12)

The second equilibrium condition means that in equilibrium, there is no net external torque to cause rotation about any axis.

The first and second equilibrium conditions are stated in a particular reference frame. The first condition involves only forces and is therefore independent of the origin of the reference frame. However, the second condition involves torque, which is defined as a cross product, ${\overrightarrow{\tau }}_k={\overrightarrow{r}}_k\times {\overrightarrow{F}}_k,$ where the position vector ${\overrightarrow{r}}_k$ with respect to the axis of rotation of the point where the force is applied enters the equation. Therefore, torque depends on the location of the axis in the reference frame. However, when rotational and translational equilibrium conditions hold simultaneously in one frame of reference, then they also hold in any other inertial frame of reference, so that the net torque about any axis of rotation is still zero. The explanation for this is fairly straightforward.

Suppose vector $\overrightarrow{R}$ is the position of the origin of a new inertial frame of reference $S\prime$ in the old inertial frame of reference S . From our study of relative motion, we know that in the new frame of reference $S\prime ,$ the position vector ${{\overrightarrow{r}}^{\prime }}_k$ of the point where the force ${\overrightarrow{F}}_k$ is applied is related to ${\overrightarrow{r}}_k$ via the equation

Now, we can sum all torques ${{\overrightarrow{\tau }}^{\prime }}_k={{\overrightarrow{r}}^{\prime }}_k\times {\overrightarrow{F}}_k$ of all external forces in a new reference frame, $S\prime :$

In the final step in this chain of reasoning, we used the fact that in equilibrium in the old frame of reference, S , the first term vanishes because of [link] and the second term vanishes because of [link] . Hence, we see that the net torque in any inertial frame of reference $S\prime$ is zero, provided that both conditions for equilibrium hold in an inertial frame of reference S .

The practical implication of this is that when applying equilibrium conditions for a rigid body, we are free to choose any point as the origin of the reference frame. Our choice of reference frame is dictated by the physical specifics of the problem we are solving. In one frame of reference, the mathematical form of the equilibrium conditions may be quite complicated, whereas in another frame, the same conditions may have a simpler mathematical form that is easy to solve. The origin of a selected frame of reference is called the pivot point .

In the most general case, equilibrium conditions are expressed by the six scalar equations ( [link] and [link] ). For planar equilibrium problems with rotation about a fixed axis, which we consider in this chapter, we can reduce the number of equations to three. The standard procedure is to adopt a frame of reference where the z -axis is the axis of rotation. With this choice of axis, the net torque has only a z -component, all forces that have non-zero torques lie in the xy -plane, and therefore contributions to the net torque come from only the x - and y -components of external forces. Thus, for planar problems with the axis of rotation perpendicular to the xy -plane, we have the following three equilibrium conditions for forces and torques: