Problem 25 - Master Resource Book in JEE Main Physics

Question

The SI unit of length is metre. Suppose we adopt a new unit of length which equal \(x\) metre. The area of \(1 \mathrm{~m}^{2}\) expressed in terms of the new unit has a magnitude (a) \(x\) (b) \(x^{2}\) (c) \(x^{-1}\) (d) \(x^{-2}\)

Step-by-Step Solution

Answer

The area in the new unit is \(x^{-2}\). Option (d).

Step 1: Understanding the Problem

We need to express the area of 1 square meter in terms of a new unit of length, where this new unit measures "\(x\) meters." We need to determine how the area changes when the unit of length changes.

Step 2: Relating Units

The standard area unit, 1 \(m^2\), is based on 1 meter by 1 meter. If each meter is now measured in terms of "\(x\) meters," then each side of the square becomes "1/x" in terms of the new unit.

Step 3: Calculating the New Area

The area of the square can be determined by multiplying the length by the width. Therefore, the area in terms of the new unit is \((1/x)\times (1/x)\), which simplifies to \(1/x^2\).

Step 4: Identifying the Correct Option

The correct option that expresses the area \(1 m^2\) in terms of the new unit is \(x^{-2}\). Comparing with the given options, the correct choice is option (d) \(x^{-2}\).

Key Concepts

Unit Conversion

When measuring physical quantities, we often need to convert between different units. This is known as unit conversion. Unit conversion allows us to express measurements in terms of a different unit system or scale. In the context of this problem, we're dealing with length measurements in meters and a new unit of length that is equivalent to a specified number of meters, say "\( x \) meters." To convert from meters to our new unit, we consider how many times this new unit fits into a standard measurement. For example, if 1 new unit equals \( x \) meters, then every measurement in meters needs to be divided by \( x \) to convert it to the new unit scale. Understanding this principle is crucial for accurate conversion and lays the foundation for solving the problem of expressing areas in new units, especially when scaling up measurements beyond simple length.

Area Measurement

Measuring area involves determining the amount of space within a two-dimensional boundary. In the standard International System of Units (SI), area is calculated using units like square meters \( m^2 \). When you're dealing with a square whose sides are each one meter long, the area is \( 1 \times 1 \) or simply \( 1 m^2 \). This calculation becomes more interesting when we change our unit of measurement. Each time we change the base unit of length, the method for calculating area must also adjust accordingly. Thus, when your dimension unit changes from meters to a new scale such as "\( x \) meters," the area calculation also changes. This occurs because the length and width transform to a new measurement dimension, which we must then multiply to find the correct area in the new unit.

New Units of Measurement

Occasionally, particular scenarios call for adopting new units of measurement, as seen in this exercise. A new unit may be defined for convenience or to better suit specific needs, for instance, when very large or very small measurements are involved. New units can be related to existing ones by defining them in terms of the base units. Here, each new unit equals \( x \) meters. Applying this to area measurement, a 1-meter side, now represented in new units, becomes \( 1/x \). Squaring this to convert to area results in \((1/x)^2 = 1/x^2\). Thus, the area of \(1 m^2\) is expressed as \(x^{-2}\) when using the new units. This approach to defining and converting new units helps maintain consistency in calculations across different measurement scales.