Animated Solution for Physics - Properties of Solids and Liquids: A 20 cm long capillary tube is dipped in water. The water rises upto 8 cm. If the entire arrangement is put in a freely falling elevator, the length of water column in the capillary tube will be

A Capillary in an Elevator

Imagine you are standing inside an elevator. You have a beaker filled with water, and you dip a $20\text{ cm}$ long capillary tube into it. Under normal, stationary conditions, the surface tension of the water pulls it up the tube against the force of gravity. The problem tells us that this initial capillary rise is $h=8\text{ cm}$.

The physics behind this is governed by the famous ascent formula:

Here, $T$ is the surface tension, $\theta$ is the angle of contact, $\rho$ is the density of the liquid, $r$ is the radius of the capillary, and $g_{\text{eff}}$ is the effective acceleration due to gravity experienced by the liquid.

Now, the situation gets thrilling. The elevator cable snaps, and the entire arrangement goes into a state of free fall.

What does free fall mean for our equations? It means the elevator is accelerating downwards with an acceleration $a=g$. Because the beaker and the capillary tube are inside this non-inertial frame of reference, we must account for a pseudo force acting upwards. This makes the effective acceleration due to gravity:

Let's substitute $g_{\text{eff}}=0$ back into our ascent formula. If the denominator approaches zero, the theoretical height $h$ approaches infinity ($h\to \infty$).

Does this mean the water will shoot out of the capillary tube like a fountain? Absolutely not. This is a classic trap in physics. A fundamental principle of fluid mechanics states that a liquid will never overflow from a capillary tube of insufficient length.

Instead of overflowing, the water will rise all the way to the very top edge of the tube. Once it reaches the top, it cannot go any further. To maintain equilibrium and satisfy the mathematical equation where $h$ is restricted to $20\text{ cm}$, the liquid dynamically adjusts its angle of contact $\theta$. The meniscus flattens out, increasing $\theta$ towards $90^{\circ }$, which makes $\cos \theta$ approach zero, perfectly balancing the zero in the denominator.

Therefore, the water column will occupy the entire available length of the tube. The final length of the water column is exactly $20\text{ cm}$.