Problem Analysis: Understanding Measurement Precision

The question asks us to identify the most precise measurement among the given options, all of which were made using a meter scale with a least count of 1 mm. Precision in measurements relates to the level of detail or the smallest unit to which a measurement is made or reported. The least count of an instrument sets the limit on the precision of measurements made with it.

The least count of the meter scale is given as 1 mm. This means the smallest division on the scale is 1 mm. When using such a scale, measurements can typically be read directly to the nearest 1 mm. It is also common practice to estimate between the smallest divisions, allowing for readings to be made with a precision of half the least count. Therefore, the absolute uncertainty in measurements made with this scale can be considered to be around $\pm 0.5$ mm.

Converting Measurements to a Common Unit

To compare the precision of measurements given in different units (mm, cm, m), it is helpful to convert them all to a single common unit. Let's convert all measurements to millimeters (mm), as the least count is given in mm.

• Option 1: $0.50 \text{ mm}$ (already in mm)
• Option 2: $29.07 \text{ cm}$. Since $1 \text{ cm} = 10 \text{ mm}$, $29.07 \text{ cm} = 29.07 \times 10 \text{ mm} = 290.7 \text{ mm}$.
• Option 3: $0.925 \text{ m}$. Since $1 \text{ m} = 1000 \text{ mm}$, $0.925 \text{ m} = 0.925 \times 1000 \text{ mm} = 925 \text{ mm}$.
• Option 4: $910 \text{ mm}$ (already in mm)

Calculating Relative Precision for Each Measurement

Precision can be assessed in different ways. One common way, especially when comparing measurements of different magnitudes made with the same instrument, is to consider the relative precision. Relative precision is calculated as the ratio of the absolute uncertainty to the measured value.

Using the least count, the absolute uncertainty ($\Delta L$) can be taken as half of the least count, which is $1 \text{ mm} / 2 = 0.5 \text{ mm}$.

The formula for relative precision is:

$$ \text{Relative Precision} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} = \frac{\Delta L}{L} $$

A smaller value for relative precision indicates a more precise measurement.

Let's calculate the relative precision for each option:

• Option 1: $L_1 = 0.50 \text{ mm}$

  $$ \text{Relative Precision}_1 = \frac{0.5 \text{ mm}}{0.50 \text{ mm}} = 1 $$

  This value indicates very low relative precision, as the uncertainty is as large as the measurement itself.

• Option 2: $L_2 = 290.7 \text{ mm}$

  $$ \text{Relative Precision}_2 = \frac{0.5 \text{ mm}}{290.7 \text{ mm}} \approx 0.00172 $$

• Option 3: $L_3 = 925 \text{ mm}$

  $$ \text{Relative Precision}_3 = \frac{0.5 \text{ mm}}{925 \text{ mm}} \approx 0.00054 $$

• Option 4: $L_4 = 910 \text{ mm}$

  $$ \text{Relative Precision}_4 = \frac{0.5 \text{ mm}}{910 \text{ mm}} \approx 0.00055 $$

Comparing Relative Precision Values

Now we compare the calculated relative precision values:

• Option 1: 1
• Option 2: $\approx 0.00172$
• Option 3: $\approx 0.00054$
• Option 4: $\approx 0.00055$

To find the most precise measurement, we look for the smallest value of relative precision. Comparing the values, we see that 0.00054 (from Option 3) is the smallest value.

$$ 0.00054 < 0.00055 < 0.00172 < 1 $$

Identifying the Most Precise Measurement

The measurement with the smallest relative precision is $0.925 \text{ m}$.

Conclusion: Determining the More Precise Measurement

Based on the calculation of relative precision using an absolute uncertainty of $0.5 \text{ mm}$ (half the least count), the measurement $0.925 \text{ m}$ has the smallest relative uncertainty and is therefore considered the most precise among the given options.

Revision Table: Key Concepts in Measurement Precision

Additional Information: Significant Figures and Uncertainty

The number of significant figures in a measurement also indicates its precision. Generally, more significant figures imply higher relative precision. However, the precision is fundamentally limited by the measuring instrument's least count.

When reporting measurements, the last significant digit is usually considered uncertain. For an instrument with a least count, the measurement is often reported to the same decimal place as the least count or one more decimal place if estimation is performed.

In this problem, interpreting "more precise" through relative precision based on the instrument's least count provides a clear method to compare measurements of different values and units.