Question

in some recent studies it has been shown that women are more susceptible to torn anterior cruciate ligaments, or acls, than men when competing in similar sports (most notably in soccer and basketball).
select the statement that explains why this disparity might exist.
the cross - sectional area of the acl is typically larger in men, and therefore experiences less strain for the same tensile force and youngs modulus.
the cross - sectional area of the acl is typically smaller in women, and therefore experiences less stress for the same tensile force.
the youngs modulus of womens acls is typically smaller than that of mens, resulting in more stress for the same amount of strain.
the acl of women is more elastic than the acl of men.

Explanation:

Step1: Recall stress - strain formula

Stress formula is \(\sigma=\frac{F}{A}\) (where \(\sigma\) is stress, \(F\) is force, \(A\) is cross - sectional area). Strain formula is \(\epsilon=\frac{\Delta L}{L_0}\). Young's modulus \(Y = \frac{\sigma}{\epsilon}\).

Step2: Analyze each option

• Option 1:

Strain \(\epsilon=\frac{\sigma}{Y}=\frac{F}{AY}\). If \(A\) (cross - sectional area) is larger for men (same \(F\) and \(Y\)), then \(\epsilon=\frac{F}{AY}\) is smaller. Smaller strain means less deformation, less likely to tear.

• Option 2:

Stress \(\sigma=\frac{F}{A}\). If \(A\) is smaller (for women, same \(F\)), then \(\sigma\) (stress) is larger (not smaller as the option says).

• Option 3:

Young's modulus \(Y=\frac{\sigma}{\epsilon}\Rightarrow\sigma = Y\epsilon\). If \(Y\) (Young's modulus) is smaller (for women), for the same \(\epsilon\) (strain), \(\sigma\) (stress) is smaller (not larger as the option says).

• Option 4:

Just saying more elastic (which is related to Young's modulus and strain - stress relationship in a more complex way than what is directly relevant here) is not as precise as the stress - strain - cross - sectional area relationship in option 1.

Answer:

The cross - sectional area of the ACL is typically larger in men, and therefore experiences less strain for the same tensile force and Young's modulus.