Problem 46 - College Physics

Question

Which of the following quantities has three significant figures: (a) \(305.0 \mathrm{~cm}\) (b) \(0.0500 \mathrm{~mm}\) (c) \(1.00081 \mathrm{~kg}\) (d) \(8.06 \times 10^{4} \mathrm{~m}^{2}\) ?

Step-by-Step Solution

Answer

Options (b) and (d) have three significant figures.

Step 1: Understand Significant Figures

Significant figures in a measurement are the digits that carry meaning contributing to its precision. This includes all non-zero digits, any zeros between significant digits, and any trailing zeros in the decimal portion.

Step 2: Evaluate Option (a)

For the measurement \(305.0\, \text{cm}\), the digits are 3, 0, and 5. The trailing zero after the decimal point indicates precision, thus the measurement has four significant figures.

Step 3: Evaluate Option (b)

The measurement \(0.0500\, \text{mm}\) begins with a zero for decimal representation, which is not significant. The significant figures are 5, 0, and the trailing zero due to decimal precision, totaling three significant figures.

Step 4: Evaluate Option (c)

In the measurement \(1.00081\, \text{kg}\), all digits are significant due to the presence of non-zero digits and zeros within the decimal portion, resulting in six significant figures.

Step 5: Evaluate Option (d)

With the measurement \(8.06 \times 10^{4}\, \text{m}^2\), both 8 and 6 are non-zero significant figures, and the zero between them is significant, resulting in three significant figures.

Key Concepts

Precision in Measurements

Precision in measurements refers to how detailed or exact a measurement is. Having more significant figures means a more precise measurement, which reduces the uncertainty in the value.

• Precision indicates the degree of refinement of a measurement.
• Different instruments offer varying levels of precision, impacting the number of significant figures you can confidently use.

To ensure measurements are precise: - Use appropriate tools. - Take multiple readings and calculate averages. Measurements with higher precision are crucial in scientific experiments, engineering, and day-to-day applications, guaranteeing consistent results and reducing errors.

Decimal Representation

Decimal representation is a way of expressing numbers that include all the digits to the right of the decimal point, showing more precision in the measurement process.

• Decimals allow us to represent fractions in a clear format.
• They help express very small or very large values.

When dealing with decimals: - Zeros in front of the decimal (such as in 0.0500) are not significant, but they help position the decimal in the correct place. - The existence of numbers after the decimal point increases the precision of a measurement, making every digit significant as long as it contributes to the value's precision.

Significant Digits

Significant digits are the numbers in a measurement that are believed to be correct, including the ones that give meaningful precision. Key points about significant digits: - All non-zero digits are significant. - Zeros between significant digits or after the decimal point can be significant. For example, consider 1.00081 kg: - Each digit here is significant because they all contribute to the precision. In summary: - Understanding significant digits helps in reporting scientific data accurately. - Correctly identifying them ensures precision and consistency across different measurements.

Trailing Zeros

Trailing zeros are zeros that appear at the end of a number. They only count as significant figures when they appear after a decimal point. Important facts about trailing zeros: - They can indicate the precision of a measurement. - For example, in the number 0.0500 mm, the zeros after '5' indicate precision. However, without decimals, as in "500," such zeros are not automatically significant unless specified otherwise, such as in scientific notation where you express it like 5.00 x 10². Introducing trailing zeros at the end of decimals helps to clarify how precise a measurement truly is, aiding in both clarity and scientific communication.