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

### Step 1: Finding the Frequency of the Tuning Fork

1. **Understanding the Beat Frequency**:
- When the wire is under a tension of 100 N, it vibrates at a frequency \( f_1 \) which is related to the tuning fork frequency \( f \) by the beat frequency of 5 beats/s. Thus, we can write:
\[
f_1 = f + 5
...