Question

The Achilles tendon connects the muscles in your calf to the back of your foot. When you are sprinting, your Achilles tendon alternately stretches, as you bring your weight down onto your forward foot, and contracts to push you off the ground. A 70 kg runner has an Achilles tendon that is 15 cm long and has a cross-section area of 110 mm2 typical values for a person of this size. Part A By how much will the runner's Achilles tendon stretch if theforce on it is 8.0 times his weight? Young's modulus for tendor is 0.15*1010N/m2. Express your answer to two significant figures and include the appropriate units. Part B What fraction of the tendon's length does this correspond. Please show ALL work.

Answer 1

The runner's Achilles tendon will stretch by approximately
`\(5.0 * 10^4 \, \text{m}\)` when subjected to a force 8.0 times the runner's weight. This corresponds to about
`\(3.3 * 10^5\)` times the tendon's original length.

To calculate the stretch of the Achilles tendon, we can use Hooke's Law, which states that the force applied to a spring or elastic material is proportional to the resulting displacement.

Hooke's Law is given by the formula:

`\[ F = (\Delta L)/(L_0) \cdot Y \cdot A \]`

where:

- F is the force applied,

-
`\( \Delta L \)` is the change in length (stretch),

-
`L_0` s the original length of the tendon,

- Y is Young's modulus,

- A is the cross-sectional area of the tendon.

Given that the force on the Achilles tendon is 8.0 times the runner's weight
`(\( 8.0 * 70 \, \text{kg} \))`, the force F is
`\( 8.0 * 70 \, \text{kg} * 9.8 \, \text{m/s}^2 \)` (acceleration due to gravity).

Let's substitute the known values into the formula and solve for
`\( \Delta L \)` in Part A:

`\[ 8.0 * 70 * 9.8 = (\Delta L)/(0.15 * 10^(10)) * 110 * 10^(-6) \]`

Now, solve for
`\( \Delta L \)`:

`\[ \Delta L = ((8.0 * 70 * 9.8) * (0.15 * 10^(10)))/(110 * 10^(-6)) \]`

Calculate the result to find
`\( \Delta L \)` and express it to two significant figures with the appropriate units.

Now, for Part B, calculate the fraction of the tendon's length that corresponds to this stretch:

`\[ \text{Fraction} = (\Delta L)/(L_0) \]`

Substitute the values for
`\( \Delta L \) and \( L_0 \)` to find the fraction and express it.

Let's continue with the calculations.

Part A:

`\[ \Delta L = ((8.0 * 70 * 9.8) * (0.15 * 10^(10)))/(110 * 10^(-6)) \]\[ \Delta L \approx (5488)/(1.1) * 10^4 \]\[ \Delta L \approx 49890 \, \text{m} \]`

Expressed to two significant figures, this is approximately
`\(5.0 * 10^4 \, \text{m}\)`.

Part B:

Now, calculate the fraction of the tendon's length:

`\[ \text{Fraction} = (\Delta L)/(L_0) \]`

Given that the original length
`(\(L_0\)) is \(15 \, \text{cm}\) (or \(0.15 \, \text{m}\))`:

`\[ \text{Fraction} = (5.0 * 10^4)/(0.15) \]\[ \text{Fraction} \approx 3.33 * 10^5 \]`

Expressed to two significant figures, this is approximately
`\(3.3 * 10^5\)`.

So, the runner's Achilles tendon would stretch by approximately
`\(5.0 * 10^4 \, \text{m}\)` and this corresponds to about
`\(3.3 * 10^5\)` times the tendon's original length.

Answer 2

Step-by-step explanation:

It is given that,

Mass of the runner, m = 70 kg

Length of the tendon, l = 15 cm = 0.15 m

Area of cross section,
`A=110\ mm^2=0.00011\ m^2`

Part A,

Let the runner's Achilles tendon stretch if the force on it is 8.0 times his weight, F = 8 mg

Young's modulus for tendon is,
`Y=0.15* 10^(10)\ N/m^2`

The formula of the Young modulus is given by :

`Y=(F/A)/((\Delta L)/(L))`

`0.15* 10^(10)=(8* 70* 9.8/0.00011)/((\Delta L)/(0.15))`

`\Delta L=0.0049\ m`

Part B,

The fraction of the tendon's length does this correspond is given by :

`(\Delta L)/(L)=(0.0049)/(0.15)`

`(\Delta L)/(L)=0.0326`

Hence, this is the required solution.