Joint Moment Arm

Although muscles produce linear forces, motions at joints are all rotary. The rotary torque is the product of the linear force and the moment arm or mechanical advantage of the muscle about the joint's center of rotation. Mechanically, this is the distance from the muscle's line of action to the joint's center of rotation.

Determination of joint moment arm requires an understanding of the anatomy and movement (kinematics) of the joint of interest. For example, some joints can be considered to rotate about a fixed point. A good example of such a joint is the elbow. At the elbow joint, where the humerus and ulna articulate, the resulting rotation occurs primarily about a fixed point, referred to as the center of rotation. In the case of the elbow joint, this center of rotation is relatively constant throughout the joint range of motion. However, in other joints (for example the knee) the center of rotation moves in space as the knee joint rotates because the articulating surfaces are not perfect circles. In the case of the knee, it is not appropriate to discuss a single center of rotation--rather we must speak of a center of rotation corresponding to a particular joint angle, or, using the terminology of joint kinematics, we must speak of the instant center of rotation (ICR), that is, the center of rotation at any "instant" in time or space.

Having defined a joint ICR, the moment arm is defined as the perpendicular distance from line of force application to the axis of rotation. This is illustrated for a simulated elbow joint. In A, the elbow joint is almost fully extended. Let the angle, q, between the brachialis muscle and the ulna be relatively small, e.g., q=20°. Let the distance between the brachialis insertion site and the elbow instant center be 5 cm. In this case, the perpendicular distance between the line of force application and the elbow ICR is shown by the dotted line in A and is equivalent to 5 cm x sin(20°) = 1.7 cm. Thus because the joint is nearly fully extended, this presents an unfavorable mechanical advantage to the muscle--the moment arm is relatively small. Much of the force generated by the muscle will simply compress the joint, not rotate it. Contrast this situation with the conditions shown in B, where the joint has now been flexed so that q=50°. Now, the moment arm equals 5 cm x sin(50°) = 4.3 cm. We see that for a simple hinge joint (a joint with a fixed ICR), the maximum moment arm is attained at q=90°. If we plotted moment arm vs. joint angle for this simple hinge joint, we would obtain a simple sine function that has a maximum of 5 cm occurring at q=90°. Such a curve can be generated for any joint. In general, the experimental curves are not quite as simple as the one here.