Problem 66 - Physics for Scientists and Engineers with Modern Physics

Question

$$ \begin{array}{l}{\text { The density of an object is defined as its mass divided by its }} \\ {\text { volume. Suppose the mass and volume of a rock are }} \\ {\text { measured to be } 8 \mathrm{g} \text { and } 2.8325 \mathrm{cm}^{3} . \text { To the correct number }} \\ {\text { of significant figures, determine the rock's density. }}\end{array} $$

Step-by-Step Solution

Answer

The density of the rock is approximately 3 g/cm³.

Step 1: Understanding Density Formula

Density is calculated using the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). In this exercise, we need to find the density of the rock using its mass and volume.

Step 2: Identifying Significant Figures

The mass of the rock is given as 8 g. The number 8 has one significant figure. The volume is given as 2.8325 cm³, which has five significant figures. The result of our calculation should be reported with the least number of significant figures, which is 1.

Step 3: Calculating the Density

Calculate the density by substituting the values into the formula:\[ \text{Density} = \frac{8 \text{ g}}{2.8325 \text{ cm}^3} \approx 2.825 \text{ g/cm}^3 \].

Step 4: Rounding the Result

Since the mass (8 g) is given to one significant figure, the calculated density should be rounded to one significant figure as well. Therefore, we round 2.825 g/cm³ to 3 g/cm³.

Key Concepts

Significant Figures

Significant figures are essential in science because they represent the precision of measured quantities. They show how many meaningful digits are in a number. **Why are they important?** They ensure that the accuracy of our calculations reflects the precision of our measurements, avoiding over-interpretation of data.

When determining significant figures in any number, consider these points:

• All non-zero numbers are always significant.
• Any zeros between significant figures are significant.
• Leading zeros are not significant, as they merely serve as placeholders.
• Trailing zeros in a decimal number are significant because they indicate precision.

Understanding how to apply significant figures in calculations is key to maintaining accuracy. In this exercise, since the mass was measured with one significant figure, the final calculated density must also reflect this level of precision.

Mass Measurement

Mass measurement is a crucial aspect in science and everyday applications. Mass represents the amount of matter in an object, often measured using a balance or scale. The unit of mass in the metric system is grams (g).

When measuring mass, it's vital to ensure the accuracy and the appropriate use of significant figures. This is especially important in scientific experiments where precise data recording is needed to achieve reliable results. For instance, in our exercise, the mass was given as 8 g, reflecting one significant figure to indicate its precision.

Any measurement should always be reported with the same level of precision as inherently measured. This practice ensures clarity in communication and interpretation of data.

Volume Measurement

Volume measures the amount of space an object occupies. It is typically expressed in cubic centimeters (cm³) or liters (L) for liquids. In scientific contexts, accurate volume measurement is as important as mass, because it influences calculations, such as density.

For objects with regular shapes, volume can be calculated using geometric formulas. However, for irregular shapes, methods like water displacement are used.

In this exercise, the volume of the rock was measured as 2.8325 cm³. This measurement has five significant figures, indicating a high degree of precision. It is essential for the volume measurement to be recorded accurately to ensure valid subsequent calculations, such as determining the density of the rock. Maintaining precision in volume measurement ensures the integrity and reliability of any derived calculations.