Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

The withdrawal resistance of a nail indicates its holding strength in wood. A formula that is used for bright common nails is \(P=15,700 S^{5 / 2} R D\), where \(P\) is the maximum withdrawal resistance (in pounds), \(S\) is the specific gravity of the wood at \(12 \%\) moisture content, \(R\) is the radius of the nail (in inches), and \(D\) is the depth (in inches) that the nail has penetrated the wood. A \(6 \mathrm{~d}\) (sixpenny) bright, common nail of length 2 inches and diameter \(0.113\) inch is driven completely into a piece of Douglas fir. If it requires a maximum force of 380 pounds to remove the nail, approximate the specific gravity of Douglas fir.

Step-by-Step Solution

Answer

The specific gravity of Douglas fir is approximately 0.542.

Step 1: Identify Given Values

From the problem statement, identify the given values:- Maximum withdrawal resistance, \( P = 380 \) pounds.- Nail diameter \( = 0.113 \) inch, so radius, \( R = \frac{0.113}{2} = 0.0565 \) inches.- Nail length (depth), \( D = 2 \) inches.

Step 2: Write Down the Formula

Recall the formula provided: \( P = 15,700 S^{5/2} R D \), where each variable is defined as in the problem statement.

Step 3: Substitute and Simplify

Substitute the known values into the formula:\[380 = 15,700 \cdot S^{5/2} \cdot 0.0565 \cdot 2\]Simplify the equation:\[380 = 15,700 \cdot 0.113 \cdot S^{5/2}\]

Step 4: Solve for Specific Gravity \(S\)

Rearrange the equation to solve for \( S^{5/2} \):\[ S^{5/2} = \frac{380}{15,700 \cdot 0.113} \]Calculate the right-hand side:\[ S^{5/2} \approx \frac{380}{1,774.1} \approx 0.214 \]

Step 5: Calculate \( S \)

To find the specific gravity \( S \), raise both sides to the power of \( \frac{2}{5} \) to yield:\[ S = (0.214)^{\frac{2}{5}} \]Calculate this:\[ S \approx 0.542 \]

Key Concepts

Specific Gravity

Specific gravity is a measure that compares the density of a substance to the density of a reference substance, typically water. It's an important factor in many fields

• In forestry and the wood industry, specific gravity helps determine mechanical properties of wood.
• It's crucial in construction, as it affects timber strength.

In this exercise, the specific gravity (\( S \),) of Douglas fir is what you are solving for. When working on problems involving specific gravity, it's key to understand that this value is dimensionless. The substance's density is divided by the density of water, making comparisons straightforward. Normal conditions for water density are approximately 1 gram per cubic centimeter, thus specific gravity above 1 means the material is denser than water, and below 1 means it's less dense.Specific gravity variations significantly affect how materials interact within environments. For wood, this property influences its strength, processing, and applications extensively.

Withdrawal Resistance

Withdrawal resistance describes the ability of a nail to resist being pulled out from wood.

• This strength largely depends on factors like the wood type, nail size, and penetration depth.
• It's essential for ensuring structural integrity in construction projects.

In the formula provided in the exercise,\( P = 15,700 S^{5/2} R D \), withdrawal resistance (\( P \),) is impacted by the specific gravity of the wood (\( S \),), the radius of the nail (\( R \),), and the depth it penetrates the wood (\( D \),).Understanding withdrawal resistance is pivotal for construction stability. If the withdrawal resistance is low, a structure may become unsound more easily since nails could fail to hold the components securely in place. Engineers and builders often depend on specific formulas and computations to ensure that designs can withstand various forces over time.

Mathematical Problem Solving

Mathematical problem-solving involves applying systematic methods to find solutions to quantitative problems.

• It's critical for understanding relationships between varying parameters through math models.
• Ensures precision and accuracy in fields requiring detailed computation, like engineering and physics.

In the exercise, your goal is to isolate the variable representing specific gravity (\( S \),) from the formula. To solve it, follow these solved steps:1. Identify the known variables: withdrawal resistance (\( P = 380 \),), nail radius (\( R \),), and depth (\( D \),).2. Substitution and simplification: Plug the known values into the equation and reduce it for manageable solving.3. Isolate the target variable (\( S^{5/2} \),) by rearranging the equation.4. Solve for \( S \), employing the inverse operation to find its actual value.Problem-solving executes the algorithmic approach by dissecting issues into manageable parts, helps anticipate potential errors or issues like incorrect assumptions, and consolidates critical thinking in resolving complex problems.