Problem 23 - Master Resource Book in JEE Main Physics

Question

The fundamental unit, which has the same power in the dimensional formulae of surface tension and viscosity is (a) mass (b) length (c) time (d) None of these

Step-by-Step Solution

Answer

(a) mass

Step 1: Determine Dimensional Formula for Surface Tension

The dimensional formula for surface tension is given by the dimensions of force per unit length. The formula is \([MLT^{-2}] \cdot [L]^{-1}\), which simplifies to \([MT^{-2}]\).

Step 2: Determine Dimensional Formula for Viscosity

Viscosity is measured as force per unit area per velocity gradient, and its dimensional formula is determined as \([FL^{-2}] \cdot [T]\) or \([M^{1}L^{-1}T^{-1}]\).

Step 3: Compare Dimensional Formulas

From Steps 1 and 2, we identify the dimensional formulas: Surface Tension \([MT^{-2}]\) and Viscosity \([M^{1}L^{-1}T^{-1}]\). The units for 'mass' \([M]\) appear with the same power in both formulas.

Key Concepts

Dimensional Formula

Dimensional analysis is a method used in physics to convert units or to understand the relationships between different physical quantities by expressing them in terms of their basic dimensions. A dimensional formula expresses the physical quantity in terms of the basic dimensions - Mass \(M\), Length \(L\), and Time \(T\). This formula helps in analyzing the proportionality and discovering consistency in equations.

To break it down further:

• The dimension of Mass is represented as \(M\).
• The dimension of Length is symbolized as \(L\).
• The dimension of Time is written as \(T\).

Understanding these basics allows us to tackle more complex derivations and validate physical laws easily. When comparing the dimensional formulas of surface tension and viscosity, we look at these basic units and their powers.

In dimensional analysis, it’s crucial to focus on the order or power of these dimensions. Identifying the same units with identical powers can bring insights, as seen with the mass dimension in surface tension and viscosity.

Surface Tension

Surface tension is a physical property of liquids where the liquid surface behaves like a stretched elastic membrane. This happens because of the cohesive forces between liquid molecules. Surface tension is the reason why small insects can walk on water or why a drop of water adopts a spherical shape.

In physics, surface tension is quantified as force per unit length, making its SI unit newtons per meter (N/m). As we learned from the dimensional analysis, the dimensional formula of surface tension is \( [MT^{-2}] \). This formula highlights the interplay of mass \(M\) and time \(T\) without including the dimension of length since it's reduced in the calculation.

Understanding surface tension is beneficial in various applications ranging from detergents, which reduce surface tension to clean clothes effectively, to essential biological processes, such as the functioning of alveoli in the lungs.

Viscosity

Viscosity is the measure of a fluid's resistance to flow. Think of it as the "thickness" or internal friction of a fluid. For instance, honey is more viscous than water. High viscosity means the fluid flows slowly, while low viscosity implies it flows more easily.

The dimensional formula for viscosity is \( [M^{1}L^{-1}T^{-1}] \). This shows how viscosity is a function of mass, inversely related to length and inversely proportional to time. It stems from the understanding of how fluids resist deformation. When you stir a liquid, you're encountering viscosity.

• Industries apply the concept of viscosity to manufacture items like lubricants and paints.
• It's also critical in processes like food processing, where controlling the viscosity of sauces and syrups is essential.

Despite viscosity and surface tension stemming from the molecular characteristics of fluids, knowing their dimensional formulas differentiates their unique behaviors clearly.