A metal wire of diameter ` 1 mm` is held on two knife by a distance ` 50 cm` . The tension in the wire is `100 N`. The wire vibrating with its fundamental frequency and `a` vibrationg tuning fork together produce ` 5 beats//s`. The tension in the wire is then reduced to `81 N`. When the two excited , beats are heard at the same rate .
Calculate
i. the frequency of the fork and
ii. the density of material of wire

Explanation

To solve the problem step by step, we will break it down into two parts: finding the frequency of the tuning fork and calculating the density of the material of the wire.

### Part (i): Calculate the frequency of the tuning fork

1. **Identify the given values:**
- Diameter of the wire, \( d = 1 \, \text{mm} = 0.001 \, \text{m} \)
- Length of the wire, \( L = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Initial tension, \( T_1 = 100 \, \text{N} \)
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