water rises in a capillary tube to a height of 1 cm. In another capillary where the radius is one-third of it, how high will the water rise (in cm)?

Answer

To solve the problem of how high water will rise in a capillary tube with a radius one-third of the original tube, we can use the formula for capillary rise. Here’s a step-by-step solution:

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

### Step 2: Identify Constants
In this problem, the liquid is the same in both capillaries, which means:
- Surface tension \( T \),
- Contact angle \( \theta \),
- Density \( \rho \),
- Acceleration due to gravity \( g \)
are all constants.

### Step 3: Relate Heights and Radii
From the formula, we can see that the height \( h \) is inversely proportional to the radius \( r \):
\[ h \propto \frac{1}{r} \]
This means:
\[ h_a \cdot r_a = h_b \cdot r_b \]
where:
- \( h_a \) is the height in the first capillary,
- \( r_a \) is the radius of the first capillary,
- \( h_b \) is the height in the second capillary,
- \( r_b \) is the radius of the second capillary.

### Step 4: Set Up the Ratios
Given:
- \( h_a = 1 \) cm (height in the first capillary),
- \( r_b = \frac{1}{3} r_a \) (radius of the second capillary is one-third of the first).

We can express the relationship as:
\[ h_a \cdot r_a = h_b \cdot r_b \]
Substituting \( r_b \):
\[ h_a \cdot r_a = h_b \cdot \left(\frac{1}{3} r_a\right) \]

### Step 5: Solve for \( h_b \)
Rearranging the equation:
\[ h_b = \frac{h_a \cdot r_a}{\frac{1}{3} r_a} \]
This simplifies to:
\[ h_b = 3 h_a \]

### Step 6: Substitute the Known Value
Now substituting \( h_a = 1 \) cm:
\[ h_b = 3 \cdot 1 \text{ cm} = 3 \text{ cm} \]

### Final Answer
The height to which water will rise in the second capillary tube is **3 cm**.

---