A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$. The angle of contact between water and the wall of the capillary tube is $\theta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

Step 1: Balance the forces acting on the water column.

The upward force due to surface tension is balanced by the weight of the water column.

$$F_{\text{up}} = 2\pi r \sigma \cos\theta$$ $$W = \rho (\pi r^2 h) g$$

Equating the upward force and the weight yields the expression for the height $h$:

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

Step 2: Analyze Statement (1) regarding the inner radius $r$.

The height $h$ is inversely proportional to the inner radius $r$.

$$h \propto \frac{1}{r}$$

As $r$ increases, $h$ decreases. Therefore, Statement (1) is true.

Step 3: Analyze Statement (2) regarding the surface tension $\sigma$.

The height $h$ is directly proportional to the surface tension $\sigma$.

$$h \propto \sigma$$

Therefore, $h$ is not independent of $\sigma$. Thus, Statement (2) is false.

Step 4: Analyze Statement (3) regarding upward acceleration.

In a lift accelerating upwards with acceleration $a$, the effective acceleration due to gravity increases.

$$g_{\text{eff}} = g + a$$ $$h = \frac{2\sigma\cos\theta}{\rho (g+a) r}$$

Since $g_{\text{eff}} > g$, the height $h$ decreases. Therefore, Statement (3) is true.

Step 5: Analyze Statement (4) regarding the contact angle $\theta$.

The height $h$ is proportional to $\cos\theta$, not directly to $\theta$.

$$h \propto \cos\theta$$

Therefore, Statement (4) is false.

Answer: Statements (1) and (3) are true.