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

Concepts and reason

The concepts used to solve this problem are the young’s modulus which is used to measure of the stiffness of a solid material.

Use the concepts of young modulus which relates stress and strain to calculate the runner’s Achilles tendon stretch and to find the ratio of a single fractional change in length over the stretched length.

Fundamentals

The expression for the weight of body is,

Here, W is the weight of the runner, m is the mass of the runner, and g is the acceleration due to gravity.

When a force is applied to an object, it will extend. Extenson of the object not only depends on the material but also on the other factors like dimensions of the object such as length, thickness etc.

Stress is defined as the force per unit area of a material.

Expression for the stress is,

Here, is the stress on the object, F is the force appiled, and A is the cross sectional area of the object.

Strain is defined as relative change in the length.

Expression for the strain is,

Here, is the strain experienced by the object, l is the stretched length, and is the original length.

Rewrite the expression for strain in terms of change in length.

Here, is the change in length.

The expression for the young’s modulus is,

(A)

Weight is gravitational force acting on a body.

Expression for the force applied runners Achilles tendon is,

Expression of young’s modulus is,

Substitute mg for F and rearrange the above equation for , the runner’s Achilles tendon stretch will be

Substitute 70 kg for m, 9.8 for g, for l, for A, and for Y.

(B)

Expression for the strain is,

Fraction of the tendon’s length is the ratio of the change in length to the stretched length.

Fraction of the tendon’s length is,

.

Substitute for and 15 cm for l to find the fraction of the tendon’s length.

Ans: Part A

Runner’s Achilles tendon stretch is.

Part B

Fraction of the tendon’s length is 0.033.

Answer 2

Part A

Runner’s Achilles tendon stretch is.

Part B

Fraction of the tendon’s length is 0.033.