CBSE Class 11 Physics Chapter 10: Mechanical Properties of Fluids NCERT Solutions

The pressure exerted by a liquid at rest is given by the hydrostatic pressure formula: $P\; =\; h\backslash rho\; g$, where $P$ is the pressure, $h$ is the height of the liquid column, $\backslash rho$ is the density of the liquid, and $g$ is the acceleration due to gravity.

a) Blood Pressure Variation:

Since pressure is directly proportional to the height ($h$) of the liquid column, the blood pressure in humans depends on the vertical distance from the heart or the difference in height between different body parts. The feet are at a lower vertical position than the brain. Therefore, the height of the blood column from the feet to the brain is greater than the height from the brain to the feet. Consequently, the blood pressure is greater at the feet compared to the brain.

b) Atmospheric Pressure Decrease with Altitude:

The density of air is not uniform throughout the atmosphere; it is highest near the Earth's surface (sea level) and decreases significantly with increasing altitude. Atmospheric pressure is largely determined by the weight of the air column above a certain point. As the density of air decreases rapidly with height, the pressure exerted by the air column also decreases. At an altitude of about 6 km, the density of air has reduced considerably, leading to the atmospheric pressure being approximately half of its value at sea level, even though the atmosphere extends much higher.

c) Hydrostatic Pressure as a Scalar:

Although pressure is defined as force per unit area ($P\; =\; F/A$), which involves force (a vector), hydrostatic pressure itself is a scalar quantity. This is because in a fluid at rest, the pressure exerted at any point is transmitted equally in all directions. Unlike a force acting on an object, which has a specific direction, the pressure within a fluid does not have a preferred direction. Therefore, hydrostatic pressure is treated as a scalar quantity.

The angle of contact ($\backslash theta$) is defined as the angle between the tangent to the liquid surface at the point of contact and the solid surface, measured inside the liquid. It depends on the relative strengths of the cohesive forces between liquid molecules and adhesive forces between liquid and solid molecules.

Let $S\_\{la\}$ be the interfacial tension between the liquid and air, $S\_\{sa\}$ be the interfacial tension between the solid and air, and $S\_\{sl\}$ be the interfacial tension between the solid and liquid. At the line of contact, the surface forces are in equilibrium. The condition for equilibrium can be expressed as:

$$S\_\{sa\}\; =\; S\_\{sl\}\; +\; S\_\{la\}\; \backslash cos\; \backslash theta$$

Rearranging this gives:

$$\backslash cos\; \backslash theta\; =\; \backslash frac\{S\_\{sa\}\; -\; S\_\{sl\}\}\{S\_\{la\}\}$$

a) Obtuse vs. Acute Angle of Contact:

For mercury with glass, the cohesive forces between mercury molecules are stronger than the adhesive forces between mercury and glass. This means $S\_\{sl\}$ is significantly smaller than $S\_\{sa\}$, leading to $S\_\{sa\}\; -\; S\_\{sl\}$ being positive but relatively small compared to $S\_\{la\}$. This results in a value of $\backslash cos\; \backslash theta$ that is positive but less than 1, and often leads to an obtuse angle ($\backslash theta\; >\; 90^\backslash circ$) because the liquid surface curves away from the solid.

For water with glass, the adhesive forces between water and glass are stronger than the cohesive forces between water molecules. This means $S\_\{sl\}$ is smaller than $S\_\{sa\}$, and $S\_\{sa\}\; -\; S\_\{sl\}$ is positive and larger. This results in a value of $\backslash cos\; \backslash theta$ that is positive and close to 1, leading to an acute angle (). The liquid surface curves upwards along the solid surface.

b) Spreading vs. Droplet Formation:

Water wets glass because the adhesive forces between water and glass are stronger than the cohesive forces within water. This strong adhesion causes the water to spread out over the glass surface to maximize contact, resulting in a small angle of contact. Conversely, mercury does not wet glass because the cohesive forces between mercury atoms are much stronger than the adhesive forces between mercury and glass. This causes the mercury to minimize its contact with the glass, forming a compact drop with an obtuse angle of contact.

c) Surface Tension and Area:

Surface tension is defined as the force acting per unit length on the surface of a liquid, perpendicular to the line separating the surface elements. It arises from the net inward pull experienced by surface molecules due to intermolecular cohesive forces. This property is an intrinsic characteristic of the liquid and depends on the intermolecular forces, not on the total area of the liquid surface. While a larger surface area requires more energy to maintain against the inward pull, the surface tension itself (force per unit length) remains constant for a given liquid under constant conditions.

d) Detergents and Angle of Contact:

Detergents reduce the surface tension of water significantly. They also alter the adhesive forces between water and the surface being cleaned. A lower surface tension allows the water to spread more easily and penetrate crevices. A smaller angle of contact, facilitated by detergents, means the liquid wets the surface better, allowing it to lift and remove dirt more effectively. Therefore, water with dissolved detergent has a smaller angle of contact.

e) Spherical Shape of Drops:

In the absence of external forces like gravity or air resistance, a liquid drop is governed by surface tension. Surface tension acts to minimize the surface area of the liquid. For a given volume, a sphere has the smallest possible surface area. Therefore, a liquid drop naturally takes on a spherical shape to minimize the surface energy associated with its surface area.