Problem 55 - Master Resource Book in JEE Main Physics

Question

Water rises to a height of \(16.3 \mathrm{~cm}\) in a capillary of height \(18 \mathrm{~cm}\) above the water level. If the tube is cut at a height of \(12 \mathrm{~cm}\) in the capillary tube, (a) water will come as a fountain from the capillary tube (b) water will stay at a height of \(12 \mathrm{~cm}\) in the capillary tube (c) the height of water in the capillary tube will be \(10.3 \mathrm{~cm}\) (d) water height flow down the sides of the capllary tube

Step-by-Step Solution

Answer

Water will stay at a height of 12 cm in the capillary tube.

Step 1: Understanding Capillary Action

Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces like gravity. In this context, water rises in a capillary tube through this phenomenon up to a maximum height determined by the properties of the liquid and the characteristics of the tube.

Step 2: Identifying Initial Conditions

The problem states that water rises to a height of 16.3 cm in a capillary tube that is 18 cm above the water level. This means under normal conditions, the water would rise to its maximum capillary height of 16.3 cm.

Step 3: Effect of Cutting the Tube

Now, the tube is cut at 12 cm, which is lower than the maximum capillary rise of 16.3 cm. Cutting the tube doesn't change the properties of the liquid or the capillary action; it only reduces the length available for the water to rise.

Step 4: Predicting Water Level After Cutting

Since the tube has been cut at 12 cm, capillary action will cause the water to rise completely to the top of the cut tube, which is 12 cm from the base, as long as this height is within the maximum capillary rise range.

Key Concepts

Surface Tension

Surface tension is a physical property that describes how the surface of a liquid behaves like a stretched elastic membrane. This occurs because the molecules in the liquid attract each other (due to forces like cohesion, which we will explain later). These attractive forces cause the surface to resist external force and maintain its shape.

One simple way to observe surface tension is by looking at how water droplets form beads on a leaf. The molecules on the liquid's surface are drawn inwards, minimizing the surface area. This tendency helps create a somewhat "tight" and "curved" surface, which is why small insects like water striders can walk on water without sinking.

Capillary action, as discussed in the original exercise, relies heavily on surface tension. When water enters a narrow tube, such as a capillary tube, the surface tension helps pull water up against gravity into the tube. This is part of the reason why water is able to rise to significant heights in capillary tubes.

Adhesion

Adhesion refers to the attraction between different types of molecules, in this case, between water molecules and the walls of the capillary tube. When water is placed into a capillary tube, the adhesion between the water molecules and the glass is a key player in drawing the liquid up the sides of the tube.

This adhesive force is crucial for capillary action. It allows the water molecules to "stick" to the walls of the tube and effectively "climb" upward, even against the force of gravity.

• Adhesion causes the surface of a liquid in a container to curve. In a narrow tube, this effect is accentuated, leading to the pronounced capillary rise.
• In everyday life, adhesion helps paint stick to walls and causes liquids to wet surfaces.

Without this adhesive attraction, the phenomenon of capillary action would not be nearly as effective as we see in nature and in the lab.

Cohesion

Cohesion is the attractive force between similar molecules, like those in water. These forces give water its characteristic "stickiness," which is why it droplets form beads on surfaces. This is due to the fact that water molecules prefer sticking to each other rather than spreading out.

Cohesive forces are responsible for the high surface tension of water. As water molecules are attracted to each other, they pull the surface tight, forming a dome-like structure or curved surfaces in drops. In the context of capillary tubes, these cohesive forces play an essential role by allowing the column of water to stay together as it rises.

While adhesion pulls water up the tube walls, cohesion keeps the body of water intact and prevents it from breaking apart. Together, adhesion and cohesion enable capillary action to work effectively, as seen when water moves up the tube until it reaches the height limited by the balance of these forces and gravity.