Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Strength of a Beam See Example \(11 .\) The strength \(S\) of a rectangular beam varies directly with its width \(W\) and the square of its thickness \(T,\) and inversely with its length \(L\). A beam that is 2 inches wide, 6 inches thick, and 96 inches long can support a load of 375 pounds. Determine how much a similar beam that is 3.5 inches wide, 8 inches thick, and 128 inches long can support.

Step-by-Step Solution

Answer

The new beam can support a load of 875 pounds.

Step 1: Understand the Direct and Inverse Variations

The strength of the beam \(S\) varies directly with its width \(W\) and the square of its thickness \(T\), which gives us \(S \propto W \cdot T^2\), and inversely with its length \(L\), thus \(S \propto \frac{1}{L}\). This combines to form the equation \(S = k \cdot \frac{W \cdot T^2}{L}\) where \(k\) is a constant.

Step 2: Use Given Data to Find the Constant

Substitute the given values of width \(W = 2\) inches, thickness \(T = 6\) inches, length \(L = 96\) inches, and strength \(S = 375\) pounds into the equation to find the constant \(k\):\[375 = k \cdot \frac{2 \cdot 6^2}{96}\]Calculate \(6^2 = 36\), substitute to have:\[375 = k \cdot \frac{2 \cdot 36}{96}\]This simplifies to:\[375 = k \cdot \frac{72}{96}\]\[375 = k \cdot \frac{3}{4}\]Solve for \(k\) by rearranging:\[k = \frac{375 \times 4}{3} = 500\]

Step 3: Determine Strength of the New Beam

Use the constant \(k = 500\) to find the strength of the new beam with \(W = 3.5\) inches, \(T = 8\) inches, and \(L = 128\) inches:\[S = 500 \cdot \frac{3.5 \cdot 8^2}{128}\]Calculate \(8^2 = 64\), substitute:\[S = 500 \cdot \frac{3.5 \cdot 64}{128}\]This reduces to:\[S = 500 \cdot \frac{224}{128}\]\[S = 500 \cdot 1.75\]Calculate the final strength:\[S = 875\]

Step 4: Final Result Interpretation

The new beam can support a load of 875 pounds, which is calculated using the derived equation and substituting the specific width, thickness, and length.

Key Concepts

Rectangular Beam Strength

Understanding the strength of a rectangular beam involves concepts from engineering and physics. The strength, denoted by \( S \), indicates how much load the beam can support before failing. The strength of a rectangular beam varies with specific dimensions: its width \( W \), the thickness \( T \), and the length \( L \).
The fundamental principle is that the strength is directly proportional to the width and the square of the thickness. This means as they increase, the strength also increases, enabling the beam to support more weight.

• Direct variation with width: The wider the beam, the more weight it can hold.
• Direct variation with thickness squared: The thicker the beam, specifically its square, greatly enhances its strength.

However, the beam strength is inversely proportional to its length. A longer beam will have decreased strength and support less load due to the increased likelihood of bending or breaking. Thus, the formula representing these relationships is:\[ S = k \cdot \frac{W \cdot T^2}{L} \]where \( k \) is a constant that characterizes the material's properties used in the beam construction.

Proportional Relationships

Proportional relationships play a crucial role in calculating beam strength. Direct and inverse proportionality influences how each dimension of the beam affects its ability to carry loads.
Direct Variation:

• Proportional to width \( W \): If you double the width, the strength also doubles, provided other factors remain the same.
• Proportional to the square of the thickness \( T^2 \): Doubling the thickness actually quadruples the strength, demonstrating a powerful impact.

Inverse Variation:

• Inversely proportional to length \( L \): Longer beams have reduced strength. If you double the length, the strength is halved.

These relationships provide the understanding needed to configure beams effectively for various uses, confirming how changes in any one dimension will impact the overall beam capability.

Physics of Materials

This area delves into how materials resist forces, ensuring that they can adequately withstand loads without failing. The interaction of different beam dimensions and material properties reveal core physics concepts:
Understanding "stress" and "strain" is essential. Stress is the internal force experienced by the material per unit area, while strain refers to the deformation due to this force.
Engineering principles tell us that a material's ability to resist applied loads depends on its inherent qualities like ductility, elasticity, and tensile strength. In beams, these properties are represented through the constant \( k \).

• Ductility indicates how much a material can deform before breaking.
• Elasticity describes a material's ability to return to its original shape after being deformed.
• Tensile strength shows the maximum stress a material can withstand while being stretched or pulled.

In practical applications, understanding these properties ensures that the chosen beam material can support the expected loads, contributing to safe and effective design in structural engineering.