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Calculate the change in length of the upper leg bone (the femur) when a 70.0 kg man supports 62.0 kg of his mass on it, assuming the bone to be equivalent to a uniform rod that is 45.0 cm long and 2.00 cm in radius.

Force of supported weight calculation

Strategy

The force is equal to the weight supported:

2. Force (Newton's second law)
FForce (N)
mMass (kg)
aAcceleration (m/s2)
Variables...
  valueunitslink
F
m
a
Bone cross-sectional area calculation

and the cross-sectional area of the upper leg bone(femur) is:

11. Disk area
AArea (m2)
πpi
RRadius (m)
Variables...
  valueunitslink
A
R
Bone length change calculation

To find the change in length we use the Young’s modulus formula. The Young’s modulus reference value for a bone under compression is known to be 9×109 N/m2. Now,all quantities except ΔL are known. Thus:

1. Young's Modulus
EYoung's modulus (modulus of elasticity) (pascal)
Fforce exerted on an object under tension (N)
L0original length of the object (m)
A0original cross-sectional area through which the force is applied (m2)
ΔLamount by which the length of the object changes (m)
Variables...
  valueunitslink
E
F
L0
A0
ΔL

Discussion

This small change in length seems reasonable, consistent with our experience that bones are rigid. In fact, even the rather large forces encountered during strenuous physical activity do not compress or bend bones by large amounts. Although bone is rigid compared with fat or muscle, several of the substances listed in Table 5.3(see reference below) have larger values of Young’s modulus Y . In other words, they are more rigid.

Reference:
This worksheet is a modified version of Example 5.4 page 188 found in :
OpenStax College,College Physics. OpenStax College. 21 June 2012.
http://openstaxcollege.org/textbooks/college-physics
Creative Commons License : http://creativecommons.org/licenses/by/3.0/

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