Respuesta :
A) We know that the gravitational Weight of a Body is given by,
[tex]F=8mg[/tex]
The expression of young's modulus is given by,
[tex]Y=\frac{Fl}{A\Delta l}[/tex]
Replacing the value of the force in the equation of young's modulis we have,
[tex]Y = \frac{8mg*l}{A\Delta l}[/tex]
Re-arrange the equation for the rate of l, we have,
[tex]\Delta l = \frac{8mgl}{AY}[/tex]
Our values here are giving by,
[tex]m=70kg[/tex]
[tex]g=9.8m/s^2[/tex]
[tex]l=15*10^{-2}m[/tex]
[tex]A=110*10^6m^2[/tex]
[tex]Y = 1.5*10^{10}N/m^2[/tex]
Replacing in the previous equation we have that,
[tex]\Delta l = \frac{8(70)(9.8)(15*10^{-2})}{110*10^6*1.5*{10}}[/tex]
[tex]\Delta l = 4.98*10^{-3}m[/tex]
We note that Achilles tendon stretch around 5mm
B) The strain is given by the equation,
[tex]\epsilon = \frac{\Delta l}{l}[/tex]
The fraction of the tendon's lenght is the ratio of the change in lenght to the streched lenght, that is
[tex]\alpha = \frac{\Delta l}{l}[/tex]
[tex]\alpha = \frac{5*10^{-3}}{15*10^{-2}}[/tex]
[tex]\alpha = 0.033[/tex]
The fraction of the tendon's lenght is 0.0333
Young's Modulus is the ratio of the stress to the strain. The strain of the Achilles tendon is 3.32x10⁻³.
What is Young's Modulus?
Young's Modulus is the ratio of the stress to the strain. therefore, it is given by the formula,
[tex]E = \dfrac{Stress}{Strain}\\\\E = \dfrac{\frac{P}{A}}{\frac{\dlta L}{L}}\\\\E= \dfrac{PL}{A\delta L}[/tex]
A.) We know that the mass of the runner is 70 kg, the length of the Achilles tendon is 15 cm(0.15 m), while the cross-sectional area of the Achilles tendon is 1.1x10⁻⁴ m², Also, the young's modulus of the Achilles tendon is 1.5x10¹⁰ N/m².
As given that the maximum force is 8 times his weight, therefore, the maximum force, P can be written as,
[tex]\rm Weight = mg\\P = 8 mg\\P = 8 \times 70 \times 9.81\\P =5493.6\ N[/tex]
As we know that the Young modulus can be written as,
[tex]E = \dfrac{PL}{A\delta L}\\\\\delta L = \dfrac{PL}{AE}[/tex]
Substitute the value,
[tex]\delta L = \dfrac{5493.6 \times 0.15}{1.1 \times 10^{-4}\times 1.5 \times 10^{10}}\\\\\delta L = 4.98 \times 10^{-4}\rm\ m[/tex]
B.) The fraction of the tendon’s length does this correspond to
The fraction of the tendon’s length this corresponds to is the strain caused in the Achilles tendon.
[tex]\epsilon = \dfrac{\delta L}{L}\\\\\epsilon = \dfrac{4.98 \times 10^{-4}}{0.15}\\\\\epsilon = 0.00332\\\\\epsilon =3.32 \times 10^{-3}[/tex]
Hence, the strain of the Achilles tendon is 3.32x10⁻³.
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