12. The ratio of the tensile (or compressive) strength to the density of a material is a measure of how strong the material is “pound for pound.” (a) Compare tendon (tensile strength 80.0 MPa, density 1100 kg/m3) with steel (tensile strength 0.50 GPa, density 7700 kg/m3): which is stronger “pound for pound” under tension? (b) Compare bone (compressive strength 160 MPa, density 1600 kg/m3) with concrete (compressive strength 0.40 GPa, density 2700 kg/m3): which is stronger “pound for pound” under compression?
To compare the strength "pound for pound," we need to calculate the ratio of tensile (or compressive) strength to density for each material.
(a) To compare tendon and steel under tension:
- Tendon: Tensile strength = 80.0 MPa (mega pascals), Density = 1100 kg/m^3.
- Steel: Tensile strength = 0.50 GPa (giga pascals), Density = 7700 kg/m^3.
To compare strength "pound for pound," we need to convert the units to a consistent system. Let's convert MPa to GPa, as we already have GPa for steel:
- 80.0 MPa = 0.080 GPa (since 1 GPa = 1000 MPa).
Now we can calculate the ratio of strength to density for each material:
- Tendon: Ratio = Tensile strength / Density = 0.080 GPa / 1100 kg/m^3.
- Steel: Ratio = Tensile strength / Density = 0.50 GPa / 7700 kg/m^3.
(b) To compare bone and concrete under compression:
- Bone: Compressive strength = 160 MPa, Density = 1600 kg/m^3.
- Concrete: Compressive strength = 0.40 GPa, Density = 2700 kg/m^3.
We need to convert MPa to GPa, as we already have GPa for concrete:
- 160 MPa = 0.160 GPa.
Now we can calculate the ratio of strength to density for each material:
- Bone: Ratio = Compressive strength / Density = 0.160 GPa / 1600 kg/m^3.
- Concrete: Ratio = Compressive strength / Density = 0.40 GPa / 2700 kg/m^3.
By comparing the two ratios for each case, we can determine which material is stronger "pound for pound" under tension and compression.
To determine which material is stronger "pound for pound" under tension and compression, we need to compare the ratios of their tensile (or compressive) strength to their density.
(a) Tendon vs. Steel under tension:
- Tendon has a tensile strength of 80.0 MPa and a density of 1100 kg/m^3.
- Steel has a tensile strength of 0.50 GPa (which is equivalent to 500 MPa) and a density of 7700 kg/m^3.
To compare them "pound for pound" under tension, we need to calculate their respective ratios of strength to density:
- For Tendon: Tensile strength / Density = 80.0 MPa / 1100 kg/m^3 = 0.0727 MPa·m^3/kg.
- For Steel: Tensile strength / Density = 500 MPa / 7700 kg/m^3 = 0.0649 MPa·m^3/kg.
Comparing the ratios, we can see that tendon has a higher strength-to-density ratio than steel. Therefore, tendon is stronger "pound for pound" under tension compared to steel.
(b) Bone vs. Concrete under compression:
- Bone has a compressive strength of 160 MPa and a density of 1600 kg/m^3.
- Concrete has a compressive strength of 0.40 GPa (which is equivalent to 400 MPa) and a density of 2700 kg/m^3.
To compare them "pound for pound" under compression, we need to calculate their respective ratios of strength to density:
- For Bone: Compressive strength / Density = 160 MPa / 1600 kg/m^3 = 0.1 MPa·m^3/kg.
- For Concrete: Compressive strength / Density = 400 MPa / 2700 kg/m^3 = 0.1481 MPa·m^3/kg.
Comparing the ratios, we can see that concrete has a higher strength-to-density ratio than bone. Therefore, concrete is stronger "pound for pound" under compression compared to bone.