When subjected to a force of compression, the length of a bone (compression Young's modulus 9.4 x 109 N/m2, tensile Young's modulus 1.6 x 1010 N/m2) decreases by 1.5 x 10-5 m. When this same bone is subjected to a tensile force of the same magnitude, by how much does it stretch?

Question

When subjected to a force of compression, the length of a bone (compression Young's modulus 9.4 x 109 N/m2, tensile Young's modulus 1.6 x 1010 N/m2) decreases by 1.5 x 10-5 m. When this same bone is subjected to a tensile force of the same magnitude, by how much does it stretch?

Answer 1

To find out how much the bone stretches when subjected to a tensile force, we need to use Hooke's Law, which states that the extension or compression of an object is directly proportional to the force applied to it.

Hooke's Law can be written as:
F = k * delta L

Where F is the force applied, k is the spring constant (which depends on the material), and delta L is the change in length.

In this case, we have two different Young's moduli for compression and tension. The Young's modulus measures the stiffness of a material. For compression, the Young's modulus is 9.4 x 10^9 N/m^2, and for tension, it is 1.6 x 10^10 N/m^2.

Let's calculate the amount of stretch for the bone when subjected to a tensile force:

F = k * delta L

Given:
Compression Young's modulus (k_compression) = 9.4 x 10^9 N/m^2
Tensile Young's modulus (k_tension) = 1.6 x 10^10 N/m^2
Change in length for compression (delta L_compression) = -1.5 x 10^-5 m

Since we are interested in the amount of stretch, we need to find the delta L for tension (delta L_tension).

We can rearrange the equation F = k * delta L to solve for delta L:

delta L = F / k

In this case, F for tension is the same magnitude as the force for compression, so we can use the same value. Let's call it F_tension.

delta L_tension = F_tension / k_tension

Now, plug in the given values:

delta L_tension = F_tension / k_tension
delta L_tension = F_tension / (1.6 x 10^10 N/m^2)

To find F_tension, we can use the relationship that the magnitude of the force applied in tension is the same as the force in compression:

F_tension = -F_compression

Substituting this into the equation:

delta L_tension = -F_compression / (1.6 x 10^10 N/m^2)

Now, plug in the values:

delta L_tension = -(-1.5 x 10^-5 m) / (1.6 x 10^10 N/m^2)

Simplifying:

delta L_tension = 1.5 x 10^-5 m / (1.6 x 10^10 N/m^2)
delta L_tension ≈ 9.375 x 10^-16 m

Therefore, when the bone is subjected to a tensile force of the same magnitude, it stretches by approximately 9.375 x 10^-16 meters.