Problem 66 - Physics for Scientists and Engineers with Modern Physics

Question

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

Step-by-Step Solution

Answer

The density is \(3 \mathrm{~g/cm}^{3}\).

Step 1: Understand the Definition of Density

Density is defined as the mass of the object divided by its volume. The formula to calculate density is given by:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]

Step 2: Identify Given Values

We are given the mass of the rock as \(8 \mathrm{~g}\) and its volume as \(2.8325 \mathrm{~cm}^{3}\). These will be used in the density formula.

Step 3: Apply the Density Formula

Substitute the given values into the density formula:\[\text{Density} = \frac{8 \mathrm{~g}}{2.8325 \mathrm{~cm}^{3}}\]

Step 4: Perform the Division Calculation

Calculate the density by performing the division:\[\text{Density} \approx 2.825 \mathrm{~g/cm}^{3}\]

Step 5: Determine Significant Figures

The mass has 1 significant figure while the volume has 5 significant figures. The result should conform to the smallest number of significant figures, which is 1 in this case.

Step 6: Round to the Correct Significant Figures

Since the calculated density is approximately \(2.825 \mathrm{~g/cm}^{3}\), and we need to round based on the smallest number of significant figures (1), the density is rounded to \(3 \mathrm{~g/cm}^{3}\).

Key Concepts

Mass

Mass represents the amount of matter within an object. It is a fundamental property that is independent of the object's shape or the gravitational pull it experiences. In scientific experiments, mass is often measured in grams (g) or kilograms (kg). To visualize mass, imagine how heavy an object feels in your hand.

Some essential points about mass include:

• Mass remains constant, even if the object's shape changes.
• It is measured using a balance or scale.
• Mass is not the same as weight; weight depends on gravity, while mass does not.

In the example with the rock, its mass is measured as 8 grams. This measurement provides the numerical value needed when using the formula for density. Knowing the mass is crucial for any calculations related to density, as it forms the numerator of the density equation.

Volume

Volume refers to the amount of space an object occupies. It helps us understand the three-dimensional size of an object. Volume is commonly measured in cubic centimeters (cm³), liters (L), or cubic meters (m³).

There are a few key aspects to remember about volume:

• Volume can be measured by water displacement for irregular objects or by mathematical formulas for standard shapes.
• In the SI system, it is derived from length measurements, such as meters, and expressed as cubic meters.
• For liquids, volume is typically measured using graduated cylinders.

In our rock example, the volume is given as 2.8325 cm³. This is a more precise measurement with five significant figures. When determining density, understanding and correctly measuring the volume is vital to obtain an accurate calculation.

Significant Figures

Significant figures are crucial in scientific measurements, as they indicate the precision of the measurement. They represent the number of digits that are known reliably, plus a last digit that is somewhat uncertain.

Here are some important points on significant figures:

• Non-zero digits are always significant.
• Zeros between significant figures are significant.
• Leading zeros are not significant.
• Trailing zeros in a decimal number are significant.

In the density calculation of the rock, the number of significant figures is determined by the least precise measurement. The mass has 1 significant figure, while the volume has 5. Thus, the calculated result also needs to have 1 significant figure. In this example, the original computed density of 2.825 g/cm³ rounded to the correct number of significant figures is 3 g/cm³. Understanding this ensures proper accuracy in reporting scientific data.