Water level in a tub is 1m . If a capaillary tube of very small radius and of length 4m is dipped into the water, it rises in the tube upto 2m . If another capillary tube of same radius but 2m length is dipped into the water then

Options

A. water will spil out from the tube
B. water will not spill out from the tube
C. water will spill out from the tube if contact angle is greater than `pi//6`
D. water will spill out from the tube if contact angle is less than `pi//6`

Answer

To solve the problem of how water behaves in a capillary tube of length 2m, we can follow these steps:

### Step 1: Understand Capillary Action
Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces. This phenomenon occurs due to the adhesive forces between the liquid and the walls of the tube and the cohesive forces within the liquid itself.

### Step 2: Use the Capillary Rise Formula
The height to which a liquid can rise in a capillary tube is given by the formula:
\[ h = \frac{2 \gamma \cos \theta}{\rho g r} \]
where:
- \( h \) = height of the liquid column
- \( \gamma \) = surface tension of the liquid
- \( \theta \) = contact angle
- \( \rho \) = density of the liquid
- \( g \) = acceleration due to gravity
- \( r \) = radius of the tube

### Step 3: Analyze the First Tube
For the first capillary tube (length = 4m), the water rises to a height of 2m. This indicates that the height of rise is determined by the radius of the tube and the properties of water.

### Step 4: Analyze the Second Tube
For the second capillary tube (length = 2m), we need to determine if the water will spill out or not. Since the height of rise in the first tube was 2m, we can conclude that the maximum height of water rise in the second tube will also be 2m, given that the radius is the same.

### Step 5: Determine the Outcome
Since the length of the second capillary tube is 2m, which is equal to the height of water rise (2m), the water will not spill out from the tube. The water will reach the top of the tube but will not overflow.

### Conclusion
Thus, when the second capillary tube of length 2m is dipped into the water, the water will not spill out from the tube.

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