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 `70dyne cm^-1`)

Options

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

Answer

To solve the problem of finding the length of the wire that does not break the surface film of water, we can follow these steps:

### Step 1: Understand the forces acting on the wire
The wire is floating on the surface of the water due to the surface tension of the water. The weight of the wire acts downwards, while the upward force due to surface tension acts on the wire.

### Step 2: Write down the weight of the wire
The weight (W) of the wire can be calculated using the formula:
\[ W = mg \]
where:
- \( m \) is the mass of the wire (1 g = 0.001 kg),
- \( g \) is the acceleration due to gravity (approximately \( 980 \, \text{cm/s}^2 \)).

### Step 3: Calculate the weight of the wire
Substituting the values:
\[ W = 1 \, \text{g} \times 980 \, \text{cm/s}^2 = 980 \, \text{dyne} \]
(Note: 1 g = 980 dyne in CGS units.)

### Step 4: Write down the force due to surface tension
The force due to surface tension (F) acting on the wire can be expressed as:
\[ F = 2T \cdot L \]
where:
- \( T \) is the surface tension of water (given as \( 70 \, \text{dyne/cm} \)),
- \( L \) is the length of the wire that is in contact with the surface.

### Step 5: Set up the equilibrium condition
For the wire to float without sinking, the upward force due to surface tension must balance the weight of the wire:
\[ W = F \]
Thus,
\[ mg = 2T \cdot L \]

### Step 6: Rearrange the equation to find L
Rearranging the equation gives:
\[ L = \frac{mg}{2T} \]

### Step 7: Substitute the known values
Now substituting the values we have:
- \( m = 1 \, \text{g} \),
- \( g = 980 \, \text{cm/s}^2 \),
- \( T = 70 \, \text{dyne/cm} \).

Substituting these values into the equation:
\[ L = \frac{980 \, \text{dyne}}{2 \times 70 \, \text{dyne/cm}} \]

### Step 8: Calculate L
Calculating this gives:
\[ L = \frac{980}{140} = 7 \, \text{cm} \]

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