force and torque equilibrium

For a rigid body at rest, there is no pivot point since there is no rotational motion. For the purposes of calculating torques, any point within the body can be used as a mathematical pivot point.

For the types of problems we will be doing, there will be two unknown forces acting at different points within the rigid body. Our example: the pulling force exerted upward by the bicep muscle, and the pushing force exerted downward by the humerus bone at the elbow on the human forearm while it is held horizontal supporting a weight in the hand.

While the elbow serves as an obvious pivot point for the arm, the reason we choose it
as a pivot point for calculating torques is that __by this choice, one of the unknown
forces is eliminated from the torque equilibrium equation__.

The general procedure for solving for the two unknown forces is this:

- Choose the pivot point to be the point of application for one of the two unknown forces.
- Calculate the torques produced by each force relative to the chosen pivot, making all of the torques positive and dividing them up into contributions clockwise and counterclockwise (less confusing than making some of the torques negative).
- Solve for the torque of the unknown force by setting (sum of clw torques) = (sum of cclw torques).
- Solve for the unknown force by dividing torque by lever arm length.
- Construct a free-body diagram which places all of the forces (including the remaining unknown) at one point of application (just as you would do for a particle dynamics problem).
- Resolve all forces into components in four directions: left, right, up and down. Make all components positive.
- Solve separately for the horizontal and vertical components of the unknown force, using the balance of components assumption (sum of components left) = (sum of components right) and (sum of components up) = (sum of components down).
- Use trigonometry to determine the magnitude of the unknown force from its known components.

In problems involving human limbs and joints, the two forces (muscle force and joint contact force) are large (typically hundreds of Newtons), are generally opposite in direction (NOT antiparallel, though), and the bigger of the two forces points upward because it must oppose both the other force and the weight.

Common pitfalls include: inability to visualize or calculate lever-arm lengths, inability to use supplementary and complementary angles to set up calculation of force components, assuming incorrectly that the given angles are to be used in torque or force component equations.