Example 9-6 Human ACL I: Maximum Force
A study of the properties of human ACLs revealed that the ultimate strength for an ACL in a younger person is, on ­average,
3.8 * 107 Pa. A typical cross-sectional area of the ACL is 4.4 * 1025 m2. What is the force exerted on the ACL when the
tensile stress is at the maximum value that the ACL can withstand?
Set Up
The ultimate strength of an object is the
maximum tensile stress, or maximum force per
area, that the object can withstand. Use Fmax to
denote this maximum force, which is what we
want to find. We use the definition of tensile
stress from Equation 9-3 to relate the ultimate
strength to the cross-sectional area A and the
force Fmax.
Solve
Rearrange the equation for ultimate strength to
solve for Fmax. Then substitute the known values
and use 1 Pa = 1 N>m2. Remember to convert
to SI units.
Reflect
Our result shows that a single knee ligament
can withstand a sizeable force of 1.7 * 103 N.
That’s greater than the weight of the heaviest
professional players in American football, those
who play offensive tackle, who have an average
mass of about 150 kg. Note that in normal use
(walking, running, and so on) the forces that
act on the ACL are only a fraction of Fmax, the
maximum force that the ACL can withstand.
Tensile stress =
F
A
area A = 4.4 x 10–5 m2
so
Ultimate strength
= maximum tensile stress =
Fmax
F max
A
Fmax = (ultimate strength) * A
= (3.8 * 107 Pa) * (4.4 * 1025 m2)
= 13.8 * 107 N>m2 2 * 14.4 * 10-5 m2 2
= 1.7 * 103 N
Offensive tackle:
mass m = 150 kg
weight mg = 1150 kg2 19.80 m>s 2 2
= 1.5 * 103 N
Fmax