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?
1.5 *10^-5 ( 9.4*10^9 /16*10^9)
= .881 * 10^-5
= 8.81 *10^-6 m
I am surprised that it is stiffer in tension than in compression
I need solution for this Question
To solve this problem, we can use Hooke's Law, which states that the strain in a material is directly proportional to the stress applied to it. The strain is calculated using the equation:
Strain = change in length / original length
When the bone is subjected to compression, the strain is given by:
Strain_compression = change in length / original length
We are given that the change in length is 1.5 x 10^(-5) m. Let's assume the original length of the bone is L.
Strain_compression = (1.5 x 10^(-5) m) / L
The stress in compression is calculated using the equation:
Stress_compression = compression Young's modulus * Strain_compression
We are given the value of compression Young's modulus as 9.4 x 10^9 N/m^2. Substituting the values, we get:
Stress_compression = (9.4 x 10^9 N/m^2) * (1.5 x 10^(-5) m / L)
Now, when the same bone is subjected to a tensile force of the same magnitude, the strain will be the same. Let's assume the amount it stretches is ΔL.
Strain_tension = ΔL /L
The stress in tension is calculated using the equation:
Stress_tension = tensile Young's modulus * Strain_tension
We are given the value of tensile Young's modulus as 1.6 x 10^10 N/m^2. Substituting the values, we get:
Stress_tension = (1.6 x 10^10 N/m^2) * (ΔL / L)
Since the strain is the same in compression and tension, the stress in compression is equal to the stress in tension. Therefore, we can equate the two equations:
Stress_compression = Stress_tension
(9.4 x 10^9 N/m^2) * (1.5 x 10^(-5) m / L) = (1.6 x 10^10 N/m^2) * (ΔL / L)
Now, we can solve the equation for ΔL, which represents the change in length when the bone is subjected to a tensile force of the same magnitude.
ΔL = [(9.4 x 10^9 N/m^2) * (1.5 x 10^(-5) m)] / (1.6 x 10^10 N/m^2)
Simplifying the equation, we get:
ΔL ≈ 8.81 x 10^(-6) m
Therefore, the bone stretches by approximately 8.81 x 10^(-6) m when subjected to a tensile force of the same magnitude.
To find out how much the bone stretches when subjected to a tensile force, we can use Hooke's law, which states that the strain (change in length) of a material is proportional to the applied stress (force per unit area). In this case, we will use the tensile Young's modulus to calculate the stretch.
The tensile Young's modulus (Et) is given as 1.6 x 10^10 N/m^2, and the strain (change in length) when compressed is 1.5 x 10^-5 m.
The formula to calculate the stress (force per unit area) is:
Stress = Force / Area
For this bone, the area doesn't change, so the stress remains the same when compressed or stretched. We can use this fact to calculate the force applied (F) by rearranging the formula:
Force = Stress * Area
Now, we can calculate the force applied when compressed:
Force_compressed = Stress * Area
Next, we can use the tensile Young's modulus to calculate how much the bone stretches when subjected to the same magnitude of force:
Stretch = Stress / Tensile Young's modulus
Finally, we can substitute the value of the stress (force per unit area) into the equation to calculate the stretch:
Stretch = (Force_compressed / Area) / Tensile Young's modulus
By substituting the known values, we can now calculate the stretch of the bone when subjected to a tensile force.