Problem 24 - Master Resource Book in JEE Main Physics

Question

The mass and volume of a body are \(4.237 \mathrm{~g}\) and \(2.5 \mathrm{~cm}^{3}\) respectively. The density of material of the body in correct significant figures is. [NCERT] (a) \(1.6048 \mathrm{~g} \mathrm{~cm}^{-3}\) (b) \(1.69 \mathrm{~g} \mathrm{~cm}^{-3}\) (c) \(1.7 \mathrm{~g} \mathrm{~cm}^{-3}\) (d) \(1.695 \mathrm{~g} \mathrm{~cm}^{-3}\)

Step-by-Step Solution

Answer

The correct density to 2 significant figures is \(1.7 \, \mathrm{g}\,\mathrm{cm}^{-3}\), option (c).

Step 1: Understand the Formula for Density

Density is the ratio of mass to volume. The formula to calculate density is given by \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). To solve the problem, we need to substitute the given values into this formula.

Step 2: Identify Significant Figures in Given Values

The mass of the body is given as \(4.237\,\mathrm{g}\) which has 4 significant figures. The volume is given as \(2.5\,\mathrm{cm}^3\) which has 2 significant figures. The result should be rounded to the least number of significant figures present in the given values, which in this case is 2 significant figures.

Step 3: Calculate the Density

Substitute the given values into the formula: \( \text{Density} = \frac{4.237\,\mathrm{g}}{2.5\,\mathrm{cm}^{3}} = 1.6948\,\mathrm{g}\,\mathrm{cm}^{-3} \).

Step 4: Round the Result to the Correct Number of Significant Figures

Since the volume 2.5 has 2 significant figures, we must round the calculated density to 2 significant figures. Thus, \(1.6948\) rounds to \(1.7 \, \mathrm{g}\,\mathrm{cm}^{-3}\).

Step 5: Select the Correct Option

Among the given options, \(1.7 \, \mathrm{g}\,\mathrm{cm}^{-3}\) is option (c), which matches our rounded density value.

Key Concepts

Density Calculation

Density is a concept commonly used in physics to describe how much mass is contained within a given volume of a substance. To calculate the density (\( \text{Density} \)) of an object, we use the simple formula:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This equation tells us that density is the mass of the object divided by its volume. The mass should be expressed in units such as grams (\( \text{g} \)) or kilograms (\( \text{kg} \)), and volume in cubic centimeters (\( \text{cm}^3 \)) or cubic meters (\( \text{m}^3 \)), depending on the context.
When calculating density, it’s crucial to have both the mass and volume values. After dividing the mass by the volume, the result gives you the density in grams per cubic centimeter (\( \text{g} \, \text{cm}^{-3} \)) or kilograms per cubic meter (\( \text{kg} \, \text{m}^{-3} \)). In our problem, we have a mass of \( 4.237 \, \text{g} \) and a volume of \( 2.5 \, \text{cm}^{3} \). Using our formula, the calculated density is \( 1.6948 \, \text{g} \text{cm}^{-3} \) before rounding.

Significant Figure Rules

Significant figures are essential in scientific calculations as they convey the precision of measurements. Understanding how many significant figures to retain in a result requires knowing a few simple rules:

• All non-zero digits are significant.
• Zeroes between non-zero digits are significant.
• Leading zeroes are not significant.
• Trailing zeroes in a decimal are significant.

The key rule for calculations is that the result should be expressed with the same number of significant figures as the measurement with the least number in the problem.
In our exercise, the mass \( 4.237 \, \text{g} \) has 4 significant figures, while the volume \( 2.5 \, \text{cm}^3 \) has 2 significant figures. Since 2 is the smaller number, our final density result should also be rounded to 2 significant figures. After rounding \( 1.6948 \, \text{g} \text{cm}^{-3} \) to 2 significant figures, we get \( 1.7 \, \text{g} \text{cm}^{-3} \).

Unit Conversion in Physics

Unit conversion is an essential skill in physics for ensuring that calculations are consistent and accurate. Units tell us what a number represents, and converting between them allows us to compare different measurements effectively. Common conversions relate to length, mass, and time.

• Mass: grams (g) to kilograms (kg) involves dividing by 1000.
• Volume: cubic centimeters (\( \text{cm}^{3} \)) to cubic meters (\( \text{m}^{3} \)) involves dividing by 1,000,000.
• Density: depending on mass and volume units, convert appropriately to consistent units.

Always ensure that you convert all measurements to compatible units before performing calculations. This prevents errors and ensures the calculated density is in the proper unit for comparison.
In our example, the units were already consistent (\( \text{grams} \) and \( \text{cubic centimeters} \)), so no further conversions were necessary. Thus, we compute density directly, resulting in units of \( \text{g} \, \text{cm}^{-3} \), appropriate for the given measurements.