Problem 66 - Master Resource Book in JEE Main Physics

Question

Crane is British unit of volume (one crane \(=170.4742\) ). Convert crane into SI units. (a) \(0.170474 \mathrm{~m}^{3}\) (b) \(17.0474 \mathrm{~m}^{3}\) (c) \(0.00170474 \mathrm{~m}^{3}\) (d) \(1704.74 \mathrm{~m}^{3}\)

Step-by-Step Solution

Answer

(d) \(1704.74 \mathrm{~m}^3\) is the SI conversion for one crane.

Step 1: Understand the Conversion

We need to convert the unit 'crane' into cubic meters (\( m^3 \)). The problem states that one crane is equal to \( 170.4742 \ m^3 \). This means that the conversion factor we will use is \( 1 \text{ crane} = 170.4742 \ m^3 \).

Step 2: Match the SI Units

Since \( 1 \) crane is \( 170.4742 \mathrm{~m}^3 \), we can directly compare the options provided with \( 170.4742 \mathrm{~m}^3 \) to see which one represents \( 1 \) crane in cubic meters.

Step 3: Analyze Each Option

Compare each option to \( 170.4742 \mathrm{~m}^3 \):- Option (a) \(= 0.170474 \mathrm{~m}^3\) is much less than one crane.- Option (b) \(= 17.0474 \mathrm{~m}^3\) is also much less than one crane.- Option (c) \(= 0.00170474 \mathrm{~m}^3\) is significantly smaller than one crane.- Option (d) \(= 170.4742 \mathrm{~m}^3\) matches exactly one crane.

Step 4: Identify the Correct Option

The correct SI unit value for one crane is \( 170.4742 \mathrm{~m}^3 \). Comparing this with the provided options, option (d) is equal to \( 170.4742 \mathrm{~m}^3 \). Thus, option (d) represents the conversion from crane to SI units correctly.

Key Concepts

Unit of Volume

Volume is a fundamental concept in various fields, such as physics and engineering. It refers to the amount of space that an object or substance occupies. In mathematics, volume is typically measured in cubic units.
For example, the volume of a cube with sides each measuring one unit is said to be one cubic unit. Commonly, volume is measured in cubic meters (\( m^3 \)) in the International System of Units (SI). This standardization helps in ensuring consistency across different scientific disciplines.
Different objects have different formulas to calculate their volume. For example, the volume of a cylinder with radius \( r \) and height \( h \) is \( \pi r^2 h \). For a sphere, the volume is \( \frac{4}{3} \pi r^3 \). Knowing these formulas can help understand how various objects relate to their space and can be particularly useful when working with volume conversions.

British Units to SI Units

In many parts of the world, traditional British units are still used, but the SI unit system is more commonly adopted for scientific accuracy. British units like feet, inches, and gallons are often tangled with terminology such as cranes, as seen in this exercise. When converting British units to SI units, it requires a clear understanding of transfer factors.
A practical example: when dealing with lengths, 1 foot is equivalent to 0.3048 meters. Similarly, converting units of volume, like gallons to cubic meters, is necessary when trying to leverage universal SI units for precise and consistent measurement across global scales.

• One crane is equivalent to \( 170.4742 \; m^3 \).
• This exercise is particularly about converting the 'crane,' a specific British unit, to the standard SI unit cubic meters.

This process aids in harmonizing measurements and allows easy interchange of information between various systems.

Conversion Factor

Conversion factors are multipliers used to convert a quantity expressed in one unit to another unit. In the context of unit conversion, a conversion factor is the ratio between two units which allows quantities to be changed from one system of measurement to another.
Consider the exercise where the conversion factor from cranes to cubic meters was given as \( 1 \text{ crane} = 170.4742 \; m^3 \). This conversion factor is crucial for solving the problem and allows seamless translation between the two different measurement systems.
To perform a conversion, the original measurement is multiplied by the conversion factor. So, if you have a measurement in cranes, multiplying it by \( 170.4742 \; m^3 \) converts it to cubic meters, giving a standardized form that's universally understood. Using conversion factors not only simplifies calculations but ensures precision and uniformity across different measurement systems.