A suspension bridge is to be built across valley where it is known that the wind can gust at `5`s intervals .It is estimated that the speed of transverse waves along the span of the bridge would be `400` m/s . The danger of resonant motions in the bridge at its fundamental frequency would be greater if the span had a length of

Options

A. `2000m`
B. `1000m`
C. `400m`
D. `80m`

Explanation

To solve the problem, we need to determine the length of the suspension bridge that would resonate with the gusts of wind occurring every 5 seconds. Here’s a step-by-step solution:

### Step 1: Determine the frequency of the wind gusts
The wind gusts occur every 5 seconds, which means the time period (T) is 5 seconds. The frequency (f) can be calculated using the formula:
\[ f = \frac{1}{T} \]
Substituting the value of T:
\[ f = \frac{1}{5} = 0.2 \, \text{Hz} \]

### Step 2: Use the wave speed to find the wavelength
The speed of transverse waves along the bridge (v) is given as 400 m/s. The relationship between wave speed (v), frequency (f), and wavelength (λ) is given by:
\[ v = f \cdot \lambda \]
We can rearrange this to find the wavelength:
\[ \lambda = \frac{v}{f} \]
Substituting the known values:
\[ \lambda = \frac{400 \, \text{m/s}}{0.2 \, \text{Hz}} = 2000 \, \text{m} \]

### Step 3: Relate the wavelength to the length of the bridge
For a bridge vibrating at its fundamental frequency, the length (L) of the bridge is related to the wavelength by the equation:
\[ L = \frac{\lambda}{2} \]
Substituting the value of λ:
\[ L = \frac{2000 \, \text{m}}{2} = 1000 \, \text{m} \]

### Conclusion
The length of the suspension bridge that would be in danger of resonant motions due to the wind gusts is:
\[ \boxed{1000 \, \text{m}} \]

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