Question

A relaxed biceps muscle requires a force of 25.0N for an elongation of 3.0 cm; under maximum tension, the same muscle requires a force 500N for the same elongation. Find the Young's modulus for the muscle tissue under each of these conditions if the muscle can be modeled as a uniform cylinder with an initial length of 0.200 m and a cross-sectional area of 50 cm^2.

Answer 1

3.3 x10^4N/m²

6.7 x105N/m²

Explanation:

Let the young modulus of the relaxed biceps be

Y= F¹Lo/ deta L1 x A

= 25 x0.2/ 0.03* 50cm²(1m²

0.0004cm^-²)

= 3.3x10^4N/m²

But young modules of muscle under maximum tension will be

Y= F"Lo/ deta L" x A

= 500x 0.2/ 0.03* 50cm²(1m²

0.0004cm^-²)

= 6.7 x10^5N/m²

Answer 2

The Young's Modulus of the relaxed muscle and the muscle under maximum tension is 3.3×10⁴ N/m² and 6.6×10⁵ N/m² respectively.

Young's Modulus:

Assuming the biceps muscle as a uniform cylinder with an initial length of L = 0.2 m and cross-sectional area of A = 50cm² = 0.05m²

(i) For the relaxed muscle:

Force required for elongation of ΔL = 0.03m is, F = 25 N

Young's Modulus (Y) = stress/strain

Now, stress =  F/A,

and strain = ΔL/L

thus,

Y = (F/A) / (ΔL/L)

Y = FL/AΔL

Y = (25×0.2)/(0.05×0.03)

Y = 3.3×10⁴ N/m²

(ii)(i) For the muscle under maximum tension:

Force required for elongation of ΔL = 0.03m is, F = 500 N

Young's Modulus (Y) = stress/strain

Now, stress =  F/A,

and strain = ΔL/L

thus,

Y = (F/A) / (ΔL/L)

Y = FL/AΔL

Y = (500×0.2)/(0.05×0.03)

Y = 6.6×10⁵ N/m²

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