How does the frequency of a vibrating wire change when the attached load is immersed in water.

Answer

To understand how the frequency of a vibrating wire changes when the attached load is immersed in water, we can follow these steps:

### Step 1: Understand the formula for frequency
The fundamental frequency \( f \) of a vibrating wire is given by the formula:
\[ f = \frac{V}{2L} \]
where \( V \) is the velocity of the wave on the wire, and \( L \) is the length of the wire.

### Step 2: Relate wave velocity to tension and mass per unit length
The wave velocity \( V \) on the wire can be expressed in terms of the tension \( T \) and the mass per unit length \( \mu \) of the wire:
\[ V = \sqrt{\frac{T}{\mu}} \]
where \( \mu = \frac{m}{L} \) (mass per unit length).

### Step 3: Analyze the effect of immersion in water
When the load attached to the wire is immersed in water, the tension \( T \) in the wire changes due to the buoyant force acting on the load. According to Archimedes' principle, the buoyant force reduces the effective weight of the load, thus reducing the tension \( T \) in the wire.

### Step 4: Determine how frequency is affected
Since the tension \( T \) decreases when the load is immersed in water, we can see that:
\[ V = \sqrt{\frac{T}{\mu}} \]
As \( T \) decreases, the wave velocity \( V \) will also decrease. Consequently, since \( L \) remains constant, the frequency \( f \) will also decrease:
\[ f = \frac{V}{2L} \Rightarrow \text{If } V \text{ decreases, then } f \text{ decreases.} \]

### Conclusion
Thus, when the attached load is immersed in water, the frequency of the vibrating wire decreases due to the reduction in tension caused by the buoyant force.

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