A 13 cm long animal tendon was found to stretch 3.8 mm by a force of 13 N. The tendon was approximately round with an average diameter of 9.0 mm. Calculate the elastic modulus of this tendon.
URGENT
*First, calculate the area:
Area = (pi)r^2 = (π)4.5^2 = 63.6173 mm^2 = 0.0000636173 m^2
*Now, use this equation to find E, the elastic modulus:
E = (F/A) ÷ (ΔL/L0)
*Convert all length measurements to m and then plug and chug:
E = (13/0.0000636173) ÷ (0.0038/0.13) = 6 990 816.05 = 7.0 x 10^6
*Elastic modulus is measured in N per m^2, so the final answer is 7.0 x 10^6 N/m^2
Further notes:
*Found r by taking diameter and dividing by two
*F = force = 13 N, listed in the problem
*A = area, which was found above (to convert from mm to m, multiply the mm figure by 10^-3; to convert from mm^2 to m^2, multiply the mm^2 figure by 10^-6, or twice the exponent of the conversion from mm to m)
*ΔL = 3.8 mm, listed in the problem
*L0 (a.k.a. "L initial") = 13 cm, listed in the problem
Oh, and the equation for Young's modulus (the elastic modulus), E = (F/A) ÷ (ΔL/L0), is actually stress divided by strain:
Stress = F/A
Strain = ΔL/L0
To calculate the elastic modulus of the tendon, we can use Hooke's law, which states that the strain (change in length divided by the original length) is proportional to the stress (force divided by the cross-sectional area) for an elastic material.
Step 1: Calculate the strain
Strain (ε) = Change in length / Original length = 3.8 mm / 13 cm = 0.038 m / 0.13 m = 0.2923
Step 2: Calculate the cross-sectional area
Cross-sectional area (A) = π * (radius)² = π * (diameter / 2)² = π * (9.0 mm / 2)² = 63.6175 mm² = 6.36175 * 10^-4 m²
Step 3: Calculate the stress
Stress (σ) = Force / Cross-sectional area = 13 N / 6.36175 * 10^-4 m² = 2.041 * 10^4 N/m²
Step 4: Calculate the elastic modulus
Elastic modulus (E) = Stress / Strain = 2.041 * 10^4 N/m² / 0.2923 = 6.98 * 10^4 N/m²
Therefore, the elastic modulus of the tendon is approximately 6.98 * 10^4 N/m².