To determine the depth of the strongest beam that can be cut out of a cylindrical log, we need to consider the principles of solid mechanics, specifically how the dimensions of a rectangular beam affect its resistance to bending. A beam's strength in bending is directly related to its section modulus, often denoted as 'Z'.

Beam Section Modulus Explained

For a rectangular beam with a width 'b' and a depth 'd', the section modulus 'Z' is given by the formula:

$Z = \frac{bd^2}{6}$

When a rectangular beam is cut from a cylindrical log with a diameter 'D', the width 'b', depth 'd', and the log's diameter 'D' are related by the Pythagorean theorem, as they form a right-angled triangle within the circular cross-section:

$b^2 + d^2 = D^2$

From this geometric relationship, we can express the width 'b' in terms of the diameter 'D' and the depth 'd':

$b = \sqrt{D^2 - d^2}$

To find the dimensions that result in the strongest beam (i.e., the beam with the maximum section modulus), we substitute the expression for 'b' into the section modulus formula:

$Z = \frac{\sqrt{D^2 - d^2} \cdot d^2}{6}$

To maximize 'Z', we need to maximize the term $\sqrt{D^2 - d^2} \cdot d^2$. A common technique is to maximize the square of this term, which simplifies the differentiation process:

Let $f(d) = (\sqrt{D^2 - d^2} \cdot d^2)^2 = (D^2 - d^2)d^4 = D^2d^4 - d^6$.

To find the maximum value of $f(d)$, we take the derivative of $f(d)$ with respect to 'd' and set it equal to zero:

$\frac{df}{dd} = 4D^2d^3 - 6d^5 = 0$

Now, we can factor out $2d^3$ from the equation:

$2d^3(2D^2 - 3d^2) = 0$

Since the depth 'd' of a beam cannot be zero, we must have:

$2D^2 - 3d^2 = 0$

Rearranging this equation to solve for 'd':

$3d^2 = 2D^2$

$d^2 = \frac{2}{3}D^2$

Taking the square root of both sides gives us the formula for the depth of the strongest beam:

$d = D\sqrt{\frac{2}{3}}$

Strongest Beam Depth Calculation

The problem provides the diameter of the cylindrical log:

• Diameter $D = 400 \text{ mm}$

Since the options are given in centimeters, it's helpful to convert the diameter to centimeters first:

• $D = 400 \text{ mm} = 40 \text{ cm}$

Now, we use the derived formula to calculate the depth 'd' of the strongest beam:

$d = D\sqrt{\frac{2}{3}}$

Substitute the value of $D = 40 \text{ cm}$ into the formula:

$d = 40 \text{ cm} \times \sqrt{\frac{2}{3}}$

Let's calculate the value of $\sqrt{\frac{2}{3}}$:

$\sqrt{\frac{2}{3}} \approx \sqrt{0.66666667} \approx 0.81649658$

Now, perform the multiplication:

$d \approx 40 \text{ cm} \times 0.81649658$

$d \approx 32.6598632 \text{ cm}$

Rounding the result to two decimal places, the depth of the strongest beam is approximately 32.66 cm.

Beam Depth Validation

Let's compare our calculated depth with the given options to find the correct one:

Our calculated depth of 32.66 cm perfectly matches Option 1. This confirms that for a cylindrical log of 400 mm diameter, the strongest rectangular beam that can be cut from it will have a depth of approximately 32.66 cm, maximizing its resistance to bending.