An ancient building has a done of 5m radius and uniform but small thickness. The surface tension of its masionary structure is about `500 "Nm"^(-1)` . Treated as hemisphere, find the maximum load that dome can support.

Answer

To find the maximum load that the dome can support, we can use the relationship between surface tension, pressure, and force. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a dome with a radius \( r = 5 \, \text{m} \) and a surface tension \( T = 500 \, \text{N/m} \). We need to calculate the maximum load that this dome can support by treating it as a hemisphere.

### Step 2: Calculate the Surface Area of the Hemisphere
The surface area \( A \) of a hemisphere is given by the formula:
\[ A = 2\pi r^2 \]
Substituting the radius:
\[ A = 2\pi (5)^2 = 2\pi (25) = 50\pi \, \text{m}^2 \]

### Step 3: Calculate the Pressure Exerted by the Surface Tension
The pressure \( P \) exerted by the surface tension can be calculated using the formula:
\[ P = \frac{4T}{r} \]
Substituting the values of \( T \) and \( r \):
\[ P = \frac{4 \times 500}{5} = \frac{2000}{5} = 400 \, \text{N/m}^2 \]

### Step 4: Calculate the Maximum Force (Load) Supported by the Dome
The maximum force \( F \) that the dome can support is given by:
\[ F = P \times A \]
Substituting the values of \( P \) and \( A \):
\[ F = 400 \times 50\pi \]
Calculating this gives:
\[ F = 20000\pi \, \text{N} \approx 62831.85 \, \text{N} \]

### Conclusion
The maximum load that the dome can support is approximately \( 62831.85 \, \text{N} \).

---