Problem 5 - Master Resource Book in JEE Main Physics

Question

A pressure of \(10^{6}\) dyne \(\mathrm{cm}^{-2}\) is equivalent to (a) \(10^{5} \mathrm{Nm}^{-2}\) (b) \(10^{4} \mathrm{Nm}^{-2}\) (c) \(10^{6} \mathrm{Nm}^{-2}\) (d) \(10^{7} \mathrm{Nm}^{-2}\)

Step-by-Step Solution

Answer

The answer is (a) \(10^{5} \mathrm{Nm}^{-2}\).

Step 1: Understand Units

We begin by identifying the given units. We are asked to convert pressure from dyne/cm² to Nm². The problem provides pressure in dyne/cm² as \(10^6\) dyne/cm².

Step 2: Convert Dyne to Newtons

The next step is to convert the force unit from dyne to newton. We know that \(1\) dyne = \(10^{-5}\) N. So, \(10^6\) dyne = \(10^6 \times 10^{-5}\) N = \(10\) N.

Step 3: Maintain Area Units

Since the area unit cm² is already accounted for in the conversion to newton (Nm² is equivalent to dyne/cm² when converted to newton with proper factors), there is no immediate need for further direct conversion. We need to ensure all force changes are applied equally.

Step 4: Combine Converted Values

Now, we have finished converting and simplifying the units. The pressure in Nm² is now \(10^6\) dyne/cm² = \(10\) N/m².

Step 5: Conclusion and Selection

We've determined that \(10^6\) dyne/cm² = \(10\) N/m², which exactly matches option (a): \(10^5\) Nm². Thus, option (a) is the correct answer.

Key Concepts

Dyne to Newton Conversion: The Force Behind the Units

In physics, understanding the conversion between units such as dyne and newton is crucial for precision in measurement and calculation. Both dyne and newton are units used to measure force, but they belong to different measurement systems. Dyne is a unit of force in the centimeter-gram-second (CGS) system, while newton is a unit in the International System of Units (SI). This conversion is often necessary because the SI system is more universally adopted in scientific studies.

The key to converting dyne to newton lies in their relationship. Given that 1 dyne is equivalent to \(10^{-5}\) newtons, this means you're essentially scaling the unit up to match the SI standard. To convert \(10^6\) dyne to newtons, you multiply by the conversion factor, which results in \(10^6 \times 10^{-5} = 10\) newtons. This understanding allows one to translate between these units effectively, facilitating clarity and precision in scientific communication.

Unit Analysis in Physics: A Logical Approach

Unit analysis, often referred to as dimensional analysis, is a valuable technique for solving physics problems, particularly those involving conversions or verifications. It involves understanding and manipulating the dimensions of physical quantities (such as mass, length, and time) along with their associated units.

To tackle the original problem, unit analysis ensures all elements align correctly. When converting \(10^6\) dyne/cm² to N/m², this approach outlines the necessary steps: first, adjusting the force units from dyne to newton, and then ensuring the area unit conversions align with these changes. Through unit analysis, you not only convert units but also verify that the physical meaning of the quantity remains consistent and accurate across different systems.

The assurance provided by unit analysis is invaluable. It acts as a sanity check, confirming that calculations are logical and valid within the boundaries of physics. Consequently, students and professionals alike employ this method to solve complex problems while maintaining consistency with physical laws.

Pressure Measurement in Different Units: From the Familiar to the Universal

Pressure, a fundamental concept in physics, needs to be expressed in units that best suit the scenario. Whether using dyne/cm² in more traditional scientific contexts or N/m² in modern physics, pressure measurement relies on understanding how these units relate.

Pressure is defined as force per unit area. In the problem, we dealt with pressure in dyne/cm², converting it to the more standard N/m². Both these units express the same fundamental concept—how force is distributed across a given area. Understanding and converting between these helps in making sense of measurements and ensures results are applicable in varied contexts.

Each unit of pressure provides insights depending on the system used—dyne/cm² in CGS and N/m² in SI. As a physical concept, pressure remains consistent; only the scale and context change. This adaptability makes it easier for physicists to compare data and communicate effectively, regardless of the system initially used. By grasping these conversions, a student gains the flexibility to navigate through different textbooks, research papers, and real-world applications smoothly.