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Let's go through the problem step by step. Here's the given exercise:

You have the following measurements of the length \( L \) (in mm) of a mechanical part measured 10 times with a ruler with graduations of 1 mm:
\[
\begin{array}{|c|c|}
\hline
\text{No.} & L \ (\text{mm}) \\
\hline
1 & 499.5 \\
2 & 500 \\
3 & 501 \\
4 & 502 \\
5 & 501 \\
6 & 499.5 \\
7 & 501.5 \\
8 & 500 \\
9 & 501.5 \\
10 & 501 \\
\hline
\end{array}
\]

The tasks are as follows:

### 1. Determine the type uncertainty linked to the reading of each measurement.
For a ruler with graduations of 1 mm, the uncertainty \( u_r \) is generally taken as half of the smallest division, so:
\[
u_r = \frac{1}{2} \, \text{mm} = 0.5 \, \text{mm}
\]
Thus, the uncertainty for each individual measurement due to the reading precision is \( u_r = 0.5 \, \text{mm} \).

### 2. Calculate the mean value of the lengths and the standard deviation \( \sigma \).

The formula for the mean \( \bar{L} \) of a set of values \( L_i \) is:
\[
\bar{L} = \frac{1}{N} \sum_{i=1}^{N} L_i
\]
where \( N = 10 \) in this case.

The standard deviation \( \sigma \) is given by:
\[
\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (L_i - \bar{L})^2}
\]

Let me calculate these values for you.The mean value of the lengths is \( \bar{L} = 500.7 \, \text{mm} \), and the standard deviation is \( \sigma = 0.843 \, \text{mm} \) (rounded to three decimal places).

### 3. Deduce the uncertainty linked to the series of measurements.
The uncertainty of the series of measurements (denoted as \( u_s \)) is related to the standard deviation and the number of measurements \( N \). It is calculated as:
\[
u_s = \frac{\sigma}{\sqrt{N}}
\]
where \( N = 10 \).

### 4. Determine the uncertainty linked to the mean value.
This uncertainty \( u_{\bar{L}} \) is equal to the uncertainty of the series of measurements \( u_s \), so:
\[
u_{\bar{L}} = u_s
\]

Let me calculate these uncertainties next.The uncertainty linked to the series of measurements is \( u_s = 0.266 \, \text{mm} \), and the uncertainty linked to the mean value is also \( u_{\bar{L}} = 0.266 \, \text{mm} \) (both rounded to three decimal places).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How does increasing the number of measurements affect the uncertainty linked to the mean value?
2. Why is the uncertainty of the mean smaller than the standard deviation of the measurements?
3. Can we reduce the uncertainty of individual measurements, and if so, how?
4. What impact would larger measurement errors have on the final results?
5. How does the choice of measurement tool affect both the uncertainty and accuracy?

**Tip:** Always repeat measurements multiple times to reduce uncertainty and obtain more reliable averages!

On mesure 10 fois consécutivement la longueur d'une pièce mécanique à l'aide d'un mètre pliant dont les graduations sont espacées de 1 mm. On obtient les résultats suivants: 499.5, 500, 501, 502, 501, 499.5, 501.5, 500, 501.5, 501. Déterminer: 1. L'incertitude-type liée à la lecture de chaque mesurage. 2. La valeur moyenne des longueurs ainsi que l'écart-type σ. 3. L'incertitude-type liée à la série de mesure. 4. L'incertitude-type liée à la valeur moyenne.

This exercise calculates the uncertainty in measuring a mechanical part 10 times using a ruler with 1 mm graduations. Learn how to determine the mean value of lengths, standard deviation, and uncertainty linked to measurements and averages. Suitable for students in advanced high school or university-level physics or engineering.

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