Problem 43 - College Physics

Question

Determine the number of significant figures in the following measured numbers: (a) \(1.007 \mathrm{~m}\), (b) \(8.03 \mathrm{~cm}\) (c) \(16.272 \mathrm{~kg}\) (d) \(0.015 \mu\) s (microseconds).

Step-by-Step Solution

Answer

(a) 4, (b) 3, (c) 5, (d) 2

Step 1: Understanding Significant Figures

Significant figures are the digits in a number that carry meaning regarding its precision. They include all non-zero digits, zeros between significant figures, and trailing zeros in the decimal portion.

Step 2: Identifying Significant Figures in 1.007 m

In the number 1.007, all the digits are significant. The non-zero digits (1 and 7) and the zeros between them are significant. Thus, it has 4 significant figures.

Step 3: Identifying Significant Figures in 8.03 cm

For the number 8.03, both the non-zero digits (8 and 3) and the zero between them are significant. Therefore, this number has 3 significant figures.

Step 4: Identifying Significant Figures in 16.272 kg

In 16.272, all digits are non-zero and therefore significant, giving a total of 5 significant figures.

Step 5: Identifying Significant Figures in 0.015 μs

The number 0.015 has two significant digits: the '1' and '5'. The leading zeros do not count as significant figures.

Key Concepts

Precision in Measurements

When we talk about measurements, precision is a key factor. Maintaining precision ensures that the data we use is as accurate as possible. Significant figures play a pivotal role in this because they represent all the known digits in a measurement plus one estimated digit. This tells us how precise a measurement is, giving us confidence in the reproducibility and reliability of the data. Always remember that more significant figures mean higher precision. This is crucial when doing any scientific calculations or experiments. For example, measuring 16.272 kg is more precise compared to 16 kg, as it has more significant figures, meaning it provides more detail about the weight.

Non-Zero Digits Significance

Non-zero digits in a number are always significant because they are undeniably part of the value of the measurement. They tell us exactly what is being measured without any ambiguity. Whether they are at the beginning, middle, or end of a number, non-zero digits signal whole parts of a measure that are known with certainty. For example, in numbers like 1.007 m and 8.03 cm, digits 1, 7, 8, and 3 are significant. They carry the measurement's true value and are always counted when determining the total number of significant figures.

Trailing Zeros in Decimals

Trailing zeros in decimal numbers play a crucial role in indicating precision. When zeros appear after a decimal point and are located at the end of a number, they are significant because they show the exactness of the measurement. These zeros confirm the measurement isn't simply an approximation. Instead, it signifies that the value neither extends all the way to the next digit nor stops short. For example, in the number 1.007, the two zeros are considered significant because they show that the measurement is precise to four digits.

Leading Zeros Insignificance

Not all zeros in measurements hold significance. Leading zeros, for instance, are zeros that come before any non-zero digit in a decimal. They don't affect the measurement's precision or the number of significant figures. Instead, they serve solely as placeholders to show the decimal point's position. In the number 0.015 μs, the leading zeros are not considered significant. They are there just to orient us within the decimal system. Therefore, the significant figures in this case are only the '1' and '5,' which tell us the actual measured quantity.