A certain concrete will support a maximum compressive stress of 1.30 107 N/m2 before being crushed. A column of uniform width is to be made with this concrete. How tall can the column be if the stress in it at any point is not to exceed one half the maximum value? The weight density of the concrete is 2.47 104 N/m3.
Enough already, what did you get?
To determine the maximum height of the column, we need to consider the stress at any point in the column and ensure it does not exceed half the maximum compressive stress.
Let's start by identifying the relevant formulas we'll be using:
1. The weight of the column can be calculated using the formula: weight = volume × density × gravitational acceleration (W = V × ρ × g).
2. The compressive stress in the column can be calculated using the formula: stress = force / area (σ = F / A).
Given information:
- Maximum compressive stress (σ_max) = 1.30 × 10^7 N/m^2
- Weight density (ρ) = 2.47 × 10^4 N/m^3
- We need to find the maximum height of the column (h).
Step 1: Calculate the weight of the column:
Since the column is of uniform width, the cross-sectional area (A) remains constant throughout. Therefore, we can obtain the weight of the column by calculating the volume (V) and multiplying it by the weight density (ρ) and gravitational acceleration (g):
weight = V × ρ × g
Step 2: Calculate the cross-sectional area:
In order to calculate the volume and the weight of the column, we need to determine the cross-sectional area (A). To do so, we need additional information, such as the width or diameter of the column. Please provide this information if available.
Once the width or diameter is known, we can calculate the area (A) accordingly.
Step 3: Determine the volume:
The volume (V) of the column can be calculated using the following formula:
V = A × h,
where h is the height of the column.
Step 4: Calculate the weight of the column:
Substituting the calculated values for V, ρ, and g into the weight formula, you can find the weight of the column.
Step 5: Determine the maximum stress at any point in the column:
Since we want to ensure that the stress in the column does not exceed half the maximum value, we can set the stress equal to half the maximum compressive stress (σ_max / 2).
Step 6: Calculate the maximum height of the column:
We can now rearrange the stress formula to solve for the height (h):
h = (weight × 2) / (A × stress).
Substituting the calculated values for weight, A, and stress, you can calculate the maximum height of the column.
Remember to use consistent units throughout the calculations for accurate results.