Chapter 8 Statics and Torque
8.6 Forces and Torques in Muscles and Joints
Summary
- Explain the forces exerted by muscles.
- State how a bad posture causes back strain.
- Discuss the benefits of skeletal muscles attached close to joints.
- Discuss various complexities in the real system of muscles, bones, and joints.
Muscles, bones, and joints are some of the most interesting applications of statics. There are some surprises. Muscles, for example, exert far greater forces than we might think. Figure 1 shows a forearm holding a book and a schematic diagram of an analogous lever system. The schematic is a good approximation for the forearm, which looks more complicated than it is, and we can get some insight into the way typical muscle systems function by analyzing it.
Muscles can only contract, so they occur in pairs. In the arm, the biceps muscle is a flexor—that is, it closes the limb. The triceps muscle is an extensor that opens the limb. This configuration is typical of skeletal muscles, bones, and joints in humans and other vertebrates. Most skeletal muscles exert much larger forces within the body than the limbs apply to the outside world. The reason is clear once we realize that most muscles are attached to bones via tendons close to joints, causing these systems to have mechanical advantages much less than one. Viewing them as simple machines, the input force is much greater than the output force, as seen below.
Example 1: Muscles Exert Bigger Forces Than You Might Think
Calculate the force the biceps muscle must exert to hold the forearm and its load as shown in Figure 1, and compare this force with the weight of the forearm plus its load. You may take the data in the figure to be accurate to three significant figures.
Strategy
There are four forces acting on the forearm and its load (the system of interest). The magnitude of the force of the biceps is FB; that of the elbow joint is FE; that of the weights of the forearm is wa, and its load is wb.Two of these are unknown (FB and FE), so that the first condition for equilibrium cannot by itself yield FB. But if we use the second condition and choose the pivot to be at the elbow, then the torque due to FE is zero, and the only unknown becomes FB.
Solution
The torques created by the weights are clockwise relative to the pivot, while the torque created by the biceps is counterclockwise; thus, the second condition for equilibrium (net τ = 0) becomes
Note that sin θ=1 for all forces, since θ=90° for all forces. This equation can easily be solved for FB in terms of known quantities, yielding
Entering the known values gives
which yields
Now, the combined weight of the arm and its load is (6.50 kg)(9.80 m/s2)=63.7 N, so that the ratio of the force exerted by the biceps to the total weight is
Discussion
This means that the biceps muscle is exerting a force 7.38 times the weight supported.
In the above example of the biceps muscle, the angle between the forearm and upper arm is 90°. If this angle ch