Water rises to a height of 4cm in a certain capillary tube. Find the height to which water will rise in another tube whose radius is one-half of the first tube.

Explanation

To solve the problem of how high water will rise in a capillary tube with a radius that is half of the first tube, we can follow these steps:

### Step 1: Understand the formula for capillary rise
The height to which a liquid rises in a capillary tube is given by the formula:
\[ h = \frac{2T \cos \theta}{r \rho g} \]
where:
- \( h \) is the height of the liquid column,
- \( T \) is the surface tension of the liquid,
- \( \theta \) is the angle of contact,
- \( r \) is the radius of the tube,
- \( \rho \) is the density of the liquid,
- \( g \) is the acceleration due to gravity.

### Step 2: Apply the formula to the first tube
For the first tube, we have:
- Height \( h_1 = 4 \, \text{cm} \)
- Radius \( r \)

Using the formula, we can express \( h_1 \):
\[ h_1 = \frac{2T \cos \theta}{r \rho g} \]

### Step 3: Apply the formula to the second tube
For the second tube, the radius is half of the first tube:
- Radius of the second tube \( r_2 = \frac{r}{2} \)

Using the formula for the second tube, we have:
\[ h_2 = \frac{2T \cos \theta}{r_2 \rho g} = \frac{2T \cos \theta}{\left(\frac{r}{2}\right) \rho g} = \frac{4T \cos \theta}{r \rho g} \]

### Step 4: Set up the ratio of heights
Now we can set up the ratio of the heights \( h_1 \) and \( h_2 \):
\[ \frac{h_1}{h_2} = \frac{\frac{2T \cos \theta}{r \rho g}}{\frac{4T \cos \theta}{r \rho g}} = \frac{1}{2} \]

### Step 5: Solve for \( h_2 \)
We know \( h_1 = 4 \, \text{cm} \), so we can substitute this into the ratio:
\[ \frac{4}{h_2} = \frac{1}{2} \]

Cross-multiplying gives:
\[ 4 = \frac{1}{2} h_2 \]

To find \( h_2 \), multiply both sides by 2:
\[ h_2 = 8 \, \text{cm} \]

### Final Answer
The height to which water will rise in the second tube is \( 8 \, \text{cm} \).

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