A string of length `50 cm` is vibrating with a fundamental frequency of `400 Hz` in horizontal position. In the fundamental mode the maximum displacement at the middle is `2 cm` from equilibrium position and the tension in the string is `10 N`. Determine the maximum value of the vertical component of force on the end supporting.

Explanation

To solve the problem step by step, we will follow the outlined approach based on the information provided in the question.

### Step 1: Understand the Setup
We have a string of length \( L = 50 \, \text{cm} = 0.5 \, \text{m} \) vibrating in its fundamental mode with a fundamental frequency \( f = 400 \, \text{Hz} \). The maximum displacement at the center (amplitude) is \( A = 2 \, \text{cm} = 0.02 \, \text{m} \), and the tension in the string is \( T = 10 \, \text{N} \).

### Step 2: Determine the Wavelength
In the fundamental mode, the string forms one complete wave (one loop) between its fixed ends. The wavelength \( \lambda \) is given by:
\[
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