Question

What is the uncertainty in each measured number: (a) \(12.60 \mathrm{~cm}\) (b) \(12.6 \mathrm{~cm}\) (c) \(0.00000003\) in. (d) \(125 \mathrm{ft}\)

Short answer

The uncertainties for the given measurements are: (a) \(12.60 \pm 0.01 \mathrm{~cm}\) (b) \(12.6 \pm 0.1 \mathrm{~cm}\) (c) \(0.00000003 \pm 0.00000001 \mathrm{~in}\) (d) \(125 \pm 1 \mathrm{~ft}\)

Step-by-step solution

a) Finding uncertainty for 12.60 cm

In this measurement, the smallest division (least count) is given up to two decimal places (hundredth of a centimeter). Therefore, the uncertainty for this measurement is ± 0.01 cm. The measurement with uncertainty will be expressed as \(12.60 \pm 0.01 \mathrm{~cm}\).

b) Finding uncertainty for 12.6 cm

In this measurement, the smallest division (least count) is given up to one decimal place (tenth of a centimeter). Therefore, the uncertainty for this measurement is ± 0.1 cm. The measurement with uncertainty will be expressed as \(12.6 \pm 0.1 \mathrm{~cm}\).

c) Finding uncertainty for 0.00000003 in

In this measurement, the smallest division (least count) is given up to 8 decimal places. Therefore, the uncertainty for this measurement is ± 0.00000001 in. The measurement with uncertainty will be expressed as \(0.00000003 \pm 0.00000001 \mathrm{~in}\).

d) Finding uncertainty for 125 ft

In this measurement, the smallest division (least count) is given as 1 ft (no decimals). Therefore, the uncertainty for this measurement is ± 1 ft. The measurement with uncertainty will be expressed as \(125 \pm 1 \mathrm{~ft}\).

Key concepts

Least Count

The least count is the smallest increment measurable by an instrument, and it plays a vital role in understanding measurement uncertainty. It represents the finest division or smallest scale on a measuring tool. Knowing the least count helps in determining the extent of possible error in a reading. If a ruler's smallest division is 0.1 cm, that is its least count. This tells us that any measurement taken with it can be off by ±0.1 cm. For example:

• In the measurement 12.60 cm, since it is measured to two decimal places, the least count is 0.01 cm.
• In the measurement 0.00000003 in, as it extends to 8 decimal places, the least count is 0.00000001 in.

Recognizing the least count allows you to express measurements accurately with associated uncertainties.

Significant Figures

Significant figures are critical in precision and expressing the uncertainty of measurements. They denote the digits in a measurement that contribute to its accuracy, excluding any that are simply placeholders. More significant figures suggest greater precision of your measurement. There are some important rules to determine significant figures:

• All non-zero digits are significant: In 12.6 cm, all the digits (1, 2, and 6) are significant.
• Zeros between significant digits are significant: The measurement 12.60 cm includes a zero, contributing to the precision of the number.
• Leading zeros are not significant: In 0.00000003 in, the zeros before the 3 are not significant.

Combining measurements with the correct number of significant figures ensures clarity and consistency in the results. Accurate use of significant figures helps convey the reliability of the measurement.

Decimal Places

Decimal places are used in measurements to indicate precision. The number of decimal places reflects the smallest unit measured and dictates the measurement's granularity. More decimal places mean more detail in the measurement. For instance:

• The measurement 12.60 cm uses two decimal places, indicating that it's measured precisely to the hundredth of a centimeter.
• The measurement 12.6 cm has just one decimal place, showing a measurement to the tenth of a centimeter.
• Measurement of 0.00000003 in uses up to 8 decimal places, offering very high precision.

The correct number of decimal places enhances the clarity and exactness of the measurement, allowing us to gauge subtle differences in measurements.

Precision in Measurement

Precision refers to the degree to which repeated measurements under unchanged conditions yield the same results. It reflects the consistency and repeatability of measurements rather than their true or standard value. Key aspects of precision in measurement include:

• Consistency: A precise measuring tool will give close to the same results under the same conditions each time, like measuring 12.60 cm several times with the same outcome.
• Resolution: This is related to the smallest change a measurement system can detect, influenced by factors like least count and decimal places.
• Uncertainty: Precision is closely tied to uncertainty. Lower uncertainty signifies a higher precision, as shown in measurements like 0.00000003 ± 0.00000001 in.

Understanding precision helps in evaluating the quality and reliability of the data collected, ensuring accurate representation of measurements.