## Illustration 13.3: The Force and Torque for Equilibrium

A rigid rod of uniform mass is shown on top of a frictionless table as shown in Animation 1. The black circle is the location of the center of mass (position is given in meters and torque is given in newton meters). Restart.

Click Animation 2 to see the forces acting on the rod (the weight and normal force cancel each other out and are not shown; they are into and out of the page, respectively). Assume that the lengths of the force vectors are indicative of the magnitudes of the forces in newtons. If these are the only forces acting on the rod, is the rod in equilibrium?

If these are the only forces acting on the rod, then it is not in equilibrium because adding up the force vectors shows that the net force on the rod is not zero. Since the net force is not zero, the center of mass of the rod will have an acceleration, and therefore its velocity will change. In addition, adding up the torques on the rod shows that the net torque on the rod about the center of mass is not zero. The rod will have a changing angular velocity out of the page.

Suppose that we want the rod to be in equilibrium. What additional force must we apply to the rod?

Consider the conditions of static equilibrium. The net force on the rod must equal zero. If you add up all of the forces presently acting on the rod (as shown in Animation 2), the sum is not zero. Therefore, we must apply another force to the rod that is the negative of the sum of the other forces on the rod.

At what location should this additional force be applied?

To be in static equilibrium, the net torque on the rod must equal zero. Therefore, the torque due to this additional force on the rod must equal the negative of the sum of the torques of the other forces presently acting on the rod. Knowing the torque and the force needed for the rod to be in equilibrium, you can calculate the location where the force should be applied to the rod.

Now, in Animation 3 you get to add a force to get the rod into equilibrium. Adjust the magnitude and direction of the blue force vector and place it at the correct location on the beam. Then check your answer. You will see a red vector for the net force and a green calculation of the net torque (this is in the z direction, which is positive out of the page). If the net force is zero, then its vector will be zero and will not be seen. If the rod is in equilibrium, the net force vector will vanish and the torque calculation will be zero, and you calculated your answers correctly. If not, recheck your calculations, adjust the magnitude, direction, and location of the blue force vector and check your answer again.

Illustration authored by Aaron Titus.
Script authored by Aaron Titus and Mario Belloni.

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