A wire of mass `1g` is kept horizontally on the surface of water. The length of the wire that does not break the surface film is (surface tension of water is `70dynecm^-1`)

Options

A. `3cm`
B. `4cm`
C. `7cm`
D. `14cm`

Explanation

To solve the problem, we need to determine the length of the wire that can rest on the surface of water without breaking the surface tension. Here’s a step-by-step solution:

### Step 1: Understand the forces acting on the wire
The wire is resting on the surface of the water, and the weight of the wire (Mg) is balanced by the upward force due to surface tension. The surface tension acts along the length of the wire.

### Step 2: Calculate the weight of the wire
The mass of the wire is given as 1 gram. To find the weight (force due to gravity), we use the formula:
\[ \text{Weight (W)} = \text{mass} \times g \]
where \( g \) is the acceleration due to gravity. In cgs units, \( g = 980 \, \text{cm/s}^2 \).

Thus,
\[ W = 1 \, \text{g} \times 980 \, \text{cm/s}^2 = 980 \, \text{dyne} \]

### Step 3: Determine the force due to surface tension
The surface tension \( S \) of water is given as \( 70 \, \text{dyne/cm} \). The total upward force due to surface tension acting on the wire can be calculated as:
\[ \text{Force due to surface tension} = 2 \times S \times L \]
where \( L \) is the length of the wire in cm. The factor of 2 accounts for the surface tension acting on both sides of the wire.

So,
\[ \text{Force due to surface tension} = 2 \times 70 \, \text{dyne/cm} \times L = 140L \, \text{dyne} \]

### Step 4: Set up the equation
For the wire to float without breaking the surface tension, the upward force due to surface tension must equal the weight of the wire:
\[ 140L = 980 \]

### Step 5: Solve for \( L \)
To find \( L \), we rearrange the equation:
\[ L = \frac{980}{140} \]
Calculating this gives:
\[ L = 7 \, \text{cm} \]

### Conclusion
The length of the wire that does not break the surface film is \( 7 \, \text{cm} \).

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