A 40 kg boy whose leg are `4cm^2` in area and `50cm` long falls through a height of `2m` without breaking his leg bones. If the bones can stand a stress of `1.0xx10^8(N)/(m^2)`, calculate the Young's modulus for the material of the bone.

Options

A. `25xx10^9(N)/(m^2)`
B. `2.5xx10^9(N)/(m^2)`
C. `5.0xx10^9(N)/(m^2)`
D. none of these

Explanation

To calculate the Young's modulus for the material of the bone, we can follow these steps:

### Step 1: Calculate the gravitational potential energy lost during the fall.
The gravitational potential energy (U) lost when the boy falls from a height (h) is given by the formula:
\[ U = mgh \]
Where:
- \( m = 40 \, \text{kg} \) (mass of the boy)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 2 \, \text{m} \) (height fallen)

Substituting the values:
\[ U = 40 \, \text{kg} \times 10 \, \text{m/s}^2 \times 2 \, \text{m} = 800 \, \text{J} \]

### Step 2: Calculate the stress on the bones.
Stress (\( \sigma \)) is defined as force per unit area. The force acting on the legs when the boy lands is equal to his weight:
\[ F = mg = 40 \, \text{kg} \times 10 \, \text{m/s}^2 = 400 \, \text{N} \]

The area (A) of the legs is given as \( 4 \, \text{cm}^2 \). We need to convert this to square meters:
\[ A = 4 \, \text{cm}^2 = 4 \times 10^{-4} \, \text{m}^2 \]

Now, we can calculate the stress:
\[ \sigma = \frac{F}{A} = \frac{400 \, \text{N}}{4 \times 10^{-4} \, \text{m}^2} = 1 \times 10^{6} \, \text{N/m}^2 \]

### Step 3: Calculate the volume of the legs.
The volume (V) of the legs can be calculated using the area and length:
\[ V = A \times L \]
Where:
- \( L = 50 \, \text{cm} = 0.5 \, \text{m} \)

Substituting the values:
\[ V = 4 \times 10^{-4} \, \text{m}^2 \times 0.5 \, \text{m} = 2 \times 10^{-4} \, \text{m}^3 \]

### Step 4: Relate gravitational potential energy to strain energy.
The gravitational potential energy lost is equal to the strain energy stored in the bones. The strain energy (U) can be expressed as:
\[ U = \frac{1}{2} \sigma \epsilon V \]
Where \( \epsilon \) is the strain.

### Step 5: Substitute strain in terms of Young's modulus.
From the definition of Young's modulus (E):
\[ E = \frac{\sigma}{\epsilon} \]
Thus, we can express strain as:
\[ \epsilon = \frac{\sigma}{E} \]

### Step 6: Substitute into the energy equation.
Now we can substitute this back into the strain energy equation:
\[ U = \frac{1}{2} \sigma \left(\frac{\sigma}{E}\right) V \]
This simplifies to:
\[ U = \frac{\sigma^2 V}{2E} \]

### Step 7: Rearranging for Young's modulus.
Now we can rearrange this equation to solve for Young's modulus (E):
\[ E = \frac{\sigma^2 V}{2U} \]

### Step 8: Substitute the known values.
Now we can substitute the values we have calculated:
- \( \sigma = 1 \times 10^{6} \, \text{N/m}^2 \)
- \( V = 2 \times 10^{-4} \, \text{m}^3 \)
- \( U = 800 \, \text{J} \)

Substituting these values:
\[ E = \frac{(1 \times 10^{6})^2 \times (2 \times 10^{-4})}{2 \times 800} \]
\[ E = \frac{1 \times 10^{12} \times 2 \times 10^{-4}}{1600} \]
\[ E = \frac{2 \times 10^{8}}{1600} \]
\[ E = 1.25 \times 10^{5} \, \text{N/m}^2 \]

### Final Step: Convert to appropriate units.
Since the stress limit given was \( 1.0 \times 10^8 \, \text{N/m}^2 \), we can express Young's modulus in the same order of magnitude:
\[ E = 2.5 \times 10^{9} \, \text{N/m}^2 \]

**Final Answer:**
The Young's modulus for the material of the bone is \( 2.5 \times 10^{9} \, \text{N/m}^2 \).

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