The Achilles tendon, which connects the calf muscles to the heel, is the thickest and strongest tendon in the body. In extreme activities, such as sprinting, it can be subjected to forces as high as 13.0 times a person's weight. According to one set of experiments, the average area of the Achilles tendon is 78.1 mm^2, its average length is 25.0 cm, and its average Young's modulus is 1474 MPa.
How much tensile stress is required to stretch this muscle by 5.00 % of its length?
.05=F(parallel)/A /Y
F(par)=
A=
Y=
If we model the tendon as a spring, what is its force constant?
If a 75.0 kg sprinter exerts a force of 13.0 times his weight on his Achilles tendon, by how much will it stretch?
F=-kx
F=13*75*9.8=9555
k=value for above part
x=solve for
You have the formulas, and the numbers to input. Why don't you just do the calculations?
For the second question, the force constant would be F(par)/(delta L)
= A Y/L
To calculate the tensile stress required to stretch the Achilles tendon by 5.00% of its length, we can use the formula:
Stress (σ) = Force (F) / Area (A)
Given data:
- Average area of the Achilles tendon (A) = 78.1 mm^2
- Average length of the Achilles tendon (L) = 25.0 cm = 250 mm
- Tensile strain (ε) = 5.00% = 0.05 (as a decimal)
First, we need to determine the force required to produce this length change. We know that stress is equal to force divided by the cross-sectional area. Rearranging the formula, we have:
Force (F) = Stress (σ) × Area (A)
We are given the average Young's modulus (Y) as 1474 MPa, which is a measure of stiffness. Young's modulus is related to stress and strain by the equation:
Young's modulus (Y) = Stress (σ) / Strain (ε)
Rearranging the formula, we can find the stress:
Stress (σ) = Young's modulus (Y) × Strain (ε)
Now we can substitute the given values:
Stress (σ) = 1474 MPa × 0.05 = 73.7 MPa
To find the force, we use the formula:
Force (F) = Stress (σ) × Area (A)
Substituting the values:
Force (F) = 73.7 MPa × 78.1 mm^2 = 5,750 N (to the nearest whole number)
The tensile stress required to stretch the Achilles tendon by 5.00% of its length is approximately 73.7 MPa, and the force required is approximately 5,750 N.
Next, to calculate the force constant of the tendon when modeled as a spring:
Hooke's Law states that the force exerted by a spring is proportional to the displacement (stretch or compression) from its equilibrium position. The formula is:
Force (F) = -k × Δx
Where:
- k is the force constant or spring constant
- Δx is the displacement from equilibrium
We are given the force (F) as 9555 N. Rearranging the equation, we can find the force constant:
k = -F / Δx
We need to determine the displacement (Δx) when the sprinter exerts a force of 13.0 times his weight. Given:
- Mass of the sprinter (m) = 75.0 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
- Force (F) = 13.0 times weight
Force (F) = 13 × weight
Weight = mass × gravitational acceleration
Substituting the values:
Force (F) = 13 × 75.0 kg × 9.8 m/s^2
Simplifying:
Force (F) = 9555 N
Now we can calculate the force constant:
k = -F / Δx
Since the force constant is not explicitly provided in the given information, we cannot calculate its exact value without the displacement (Δx) data.
To calculate the tensile stress required to stretch the Achilles tendon by 5.00% of its length, we can use the formula:
Stress = Force / Area
Given:
Average area (A) of the Achilles tendon = 78.1 mm^2
Average length (L) of the Achilles tendon = 25.0 cm = 250 mm
Young's modulus (Y) of the Achilles tendon = 1474 MPa
First, convert the area from mm^2 to m^2:
A = 78.1 mm^2 = 78.1 * (10^-6) m^2
Next, calculate the force (F) required to stretch the tendon by 5.00% of its length:
Change in length (ΔL) = 5.00% of 250 mm = 0.05 * 250 mm = 12.5 mm = 12.5 * (10^-3) m
F = Stress * Area
F = Stress * 78.1 * (10^-6) m^2
Using the formula:
Stress = (Force / Area) / Young's Modulus
We can rearrange the formula to solve for Stress:
Stress = (F / A) / Y
Now, substitute the values into the formula:
(F / A) / Y = (Force / Area) / Young's Modulus
Since we want to find the tensile stress required to stretch the muscle by 5.00% of its length, we isolate Stress:
Stress = (F / A) / Y
To calculate the force constant of the tendon when modeled as a spring, we can use Hooke's Law:
Force (F) = -k * x
Given:
Force (F) = 13 * weight of the sprinter
Weight of the sprinter = 75.0 kg
So,
F = 13 * 75.0 * 9.8
To solve for the force constant (k), rearrange the equation:
F = -k * x
Substituting the values into the equation, we have:
13 * 75.0 * 9.8 = -k * x
Finally, solve for x, which represents the amount by which the tendon will stretch.