Problem 15 - College Physics

Question

Bone has a Young's modulus of \(18 \times 10^{9} \mathrm{~Pa}\). Under compression, it can withstand a stress of about \(160 \times 10^{6} \mathrm{~Pa}\) before breaking. Assume that a femur (thigh bone) is \(0.50 \mathrm{~m}\) long, and calculate the amount of compression this bone can withstand before breaking.

Step-by-Step Solution

Answer

The bone can withstand a compression of about 4.45 mm before breaking.

Step 1: Understand the Given Values

The Young's modulus \(E\) for bone is given as \(18 \times 10^{9} \mathrm{~Pa}\) and the stress (\(\sigma\)) it can withstand before breaking is \(160 \times 10^{6} \mathrm{~Pa}\). The length (\(L\)) of the femur is 0.50 m.

Step 2: Define the Relevant Equation

The formula to calculate strain (\(\epsilon\)), is derived from the formula of Young's Modulus, \(E = \sigma / \epsilon \), can be rearranged as \(\epsilon = \sigma / E\).

Step 3: Insert the Known values

Substitute the given values into the equation: \(\epsilon = \sigma / E = (160 \times 10^{6} \mathrm{~Pa}) / (18 \times 10^{9} \mathrm{~Pa})\).

Step 4: Calculate the Strain

By doing the division, the strain comes out to be about \(\epsilon \approx 0.0089 \).

Step 5: Calculate Compression

The amount of compression this bone can withstand before it breaks can be obtained by multiplying the total length of the bone with the strain. This yields \(0.50 \, m \times 0.0089 = 0.00445 \, m\) or 4.45 mm.

Key Concepts

Understanding Strain Calculation

Strain is a measure of deformation representing the displacement between particles in the material body. It's important to know that strain is dimensionless, meaning it has no unit. This is because strain is a ratio of lengths: the change in length divided by the original length.
To compute strain (\(\epsilon\)), you can use the formula derived from Young's Modulus:

• \(\epsilon = \frac{\sigma}{E}\)
• \(\sigma\) is the stress applied to the material
• \(E\) represents Young's Modulus

In our exercise, the known stress (\(160 \times 10^{6} \,\mathrm{Pa}\)) is inserted into the equation together with the Young's Modulus for bone (\(18 \times 10^{9} \,\mathrm{Pa}\)) to find the strain. The resultant strain was calculated to be about \(0.0089\), indicating that the bone slightly elongates under stress before returning to its original shape after the load is removed.

Conducting Stress Analysis

Stress analysis allows us to understand how a material behaves under various forces. In essence, stress (\(\sigma\)) is defined as force exerted over an area (\(F/A\)). Stress is measured in pascals (Pa) in the SI unit system. Key points to consider when working with stress analysis include:

• Always consider the type of stress (compressive, tensile, or shear)
• Be aware of the material's ability to handle stress before failure

In the example, the femur can withstand a stress of \(160 \times 10^{6} \,\mathrm{Pa}\). Knowing the bones' stress limit is crucial in preventing excess stress that can lead to fracture or failure.

Exploring Material Properties

Material properties, such as Young's Modulus, are intrinsic qualities that dictate how materials respond to forces. Young's Modulus (\(E\)) specifically measures a material's stiffness, reflecting resistance to elastic deformation. For bone, this modulus is cited as \(18 \times 10^{9} \,\mathrm{Pa}\). Higher values indicate more stiffness, and therefore, less deformation under the same stress.
Understanding these properties helps engineers and scientists predict material behaviors in structures and biological contexts. For example, for the femur, we know how much it can compress without reaching the breaking point, considering its length and the stress it can bear. This insight is invaluable for practical applications, such as designing prosthetics or surgical procedures.