A cylindrical capillary tube of $0.2\,\text{mm}$ radius is made by joining two capillaries $T_1$ and $T_2$ of different materials having water contact angles of $0^\circ$ and $60^\circ$, respectively. The capillary tube is dipped vertically in water in two different configurations, case $I$ and $II$ as shown in figure. Which of the following option(s) is (are) correct? [surface tension of water $= 0.075\,\text{N/m}$, density of water $= 1000\,\text{kg/m}^3$, take $g = 10\,\text{m/s}^2$]

Step 1: Identify the given parameters.

The parameters for the capillary tube and water are:

$$r = 0.2\,\text{mm} = 2 \times 10^{-4}\,\text{m}$$ $$\sigma = 0.075\,\text{N/m}$$ $$\rho = 1000\,\text{kg/m}^{3}$$ $$g = 10\,\text{m/s}^{2}$$

The contact angles for capillaries $T_1$ and $T_2$ are:

$$\theta_1 = 0^\circ$$ $$\theta_2 = 60^\circ$$

Step 2: Calculate the maximum capillary rise for each material.

The formula for capillary rise is:

$$h = \frac{2\sigma\cos\theta}{\rho g r}$$

For capillary $T_1$:

$$h_{T_1} = \frac{2 \times 0.075 \times \cos 0^\circ}{1000 \times 10 \times 2 \times 10^{-4}}$$ $$h_{T_1} = 0.075\,\text{m} = 7.5\,\text{cm}$$

For capillary $T_2$:

$$h_{T_2} = \frac{2 \times 0.075 \times \cos 60^\circ}{1000 \times 10 \times 2 \times 10^{-4}}$$ $$h_{T_2} = 7.5 \times 0.5 = 3.75\,\text{cm}$$

Step 3: Analyze Case II with joint height $H = 5\,\text{cm}$.

In Case II, the lower capillary is $T_2$ and the upper capillary is $T_1$. Since the joint is at $H = 5\,\text{cm}$ and the maximum rise in $T_2$ is $h_{T_2} = 3.75\,\text{cm}$:

$$h_{T_2} < H$$

The water level cannot reach the joint. Thus, the meniscus stays in $T_2$ at a height of:

$$h = 3.75\,\text{cm}$$

This statement is correct.

Step 4: Analyze Case I with joint height $H = 8\,\text{cm}$.

In Case I, the lower capillary is $T_1$ and the upper capillary is $T_2$. Since the joint is at $H = 8\,\text{cm}$ and the maximum rise in $T_1$ is $h_{T_1} = 7.5\,\text{cm}$:

$$h_{T_1} < H$$

The water level cannot reach the joint. Thus, the meniscus stays in $T_1$ at a height of:

$$h = 7.5\,\text{cm}$$

This statement is correct.

Step 5: Analyze Case I with joint height $H = 5\,\text{cm}$.

Since $H = 5\,\text{cm} < h_{T_1} = 7.5\,\text{cm}$, the water easily reaches the joint. However, the upper capillary $T_2$ can only support a maximum height of $3.75\,\text{cm}$, which is less than the joint height of $5\,\text{cm}$.

Consequently, the water cannot rise further into $T_2$, and the meniscus becomes pinned at the joint:

$$h = 5\,\text{cm}$$

Thus, the statement claiming the rise is more than $8.75\,\text{cm}$ is incorrect.

Step 6: Analyze the meniscus weight correction.

The correction to the height of the water column due to the weight of the meniscus is given by:

$$\Delta h = \frac{V_{\text{meniscus}}}{\pi r^{2}}$$

Since the contact angles $\theta_1 = 0^\circ$ and $\theta_2 = 60^\circ$ are different, the shapes and volumes of the menisci in Case I and Case II are different:

$$V(\theta = 0^\circ) \neq V(\theta = 60^\circ)$$

Therefore, the height correction is different for Case I and Case II. This statement is correct.

Answer: The correct options are:

1. The correction to the height of water column due to meniscus shape is different for Case I and Case II.
2. For Case II, if the joint is at a height of $5\,\text{cm}$, the water meniscus will stay in $T_2$ and the rise will be $3.75\,\text{cm}$.
3. For Case I, if the joint is at a height of $8\,\text{cm}$, the water meniscus will stay in $T_1$ and the rise will be $7.5\,\text{cm}$.