## Learning objectives

This module, as the whole course, aims to bring students to a level of knowledge of measurement methodologies that enable them to manage with autonomy simple laboratory experiments for the determination of mechanical quantities and calorimetry. Also aims to provide students with a basic knowledge of the theory of errors with elements of probability theory and stochastic variables.

At the end of the module the student will be able to:

Knowledge and understanding

To know the basic concepts of uncertainty, error, measurement, the normal distribution and its use. To understand the difference between frequency and probability and how to predict the probability of an event. To know the basi physical principle necessary to carry on an experiment.

Applying knowledge and understanding:

To carry on simple quantitative experiments in physics; to be able to use the basic laboratory instrumentation; to be able to perform simple statistical data processing; to be able to use the main computer tools for the data analysis.

Making judgements:

To evaluate and process the statistical and systematic errors; to evaluate the data quality according their uncertainty; to discriminate outliers; to interpret the differences between expected and obtained results.

Communication skills:

To write synthetic and complete laboratory reports; to communicate with partners in order to plan experiment and share knowledge, to correctly expose graphical and numerical data using a formally correct language.

Learning skills:

To integrate data and information coming from different sources; to plan improvements for their experiments; to test their theoretical knowledge according the experimental results.

## Prerequisites

Some basic concepts of math: algebra, trigonometry, analytic geometry, differential and integral calculus.

Some basic concepts in physics: kinematics and dynamics of material point, calorimetry.

## Course unit content

Module I

1. Measurement and uncertainty

2. Error propagation and its representation

3. Statistical treatment of data and their representation

4. Gaussian distribution

5. Further elements of data analysis

6. Introduction to probability theory

7. Short account on Combinatorics

8. Short account on Calorimetry

The Laboratory experiments will cover:

• Basic measurements of physical quantities

• Free body fall

• One-dimensional harmonic motion

• Motion of torsion pendulum

• Bernoulli and Poisson distributions

• The adiabatic calorimeter

## Full programme

Module I

1. The measurement: direct and indirect measurements of physical quantities, units, characteristics and selection criteria of measuring instruments: accuracy, precision, promptness, dynamic range. Systematic and random errors, confidence intervals; orders of magnitude and significant figures.

2. Study of uncertainties in physical measurements: error propagation (sum, difference, product, quotient, the sum in quadrature, error as a function of one and two variables), error as differential. Measurement errors and their representation: confidence interval, significant digits, consistency / discrepancy between measurements, verification of physical laws.

3. Study of uncertainties in physical measurements: statistical treatment of data and their representation; statistical analysis of random errors: mean, variance and standard deviation, histograms and frequency distributions. Cumulative frequency. Short account on the treatment of systematic errors.

4. Study of uncertainties in physical measurements: frequency and probability, the limit distribution, probability density, normalization, mean value and standard deviation. Gaussian distribution: confidence and standard deviation, standard error integral; comparison of results. Mean as the best estimate. Population distributions.

5. Study of uncertainties in physical measurements: weighted averages, data rejection (Chauvenet criterion). Short account on the method of least squares and regression.

6. Introduction to probability theory: statistics and probability, discrete and continuous variables, the concept of event; favorable and possible cases, classical and frequentist definition of probability.

7. Combinatorics: simple distributions, distributions with repetition, simple permutations, permutations with identical objects, simple combinations, combinations with repetition. Lottery games.

8. Elements of calorimetry: definition of temperature, methods of temperature measurement, thermocouples, specific heat and heat capacity. Mechanisms of heat transfer, calorimeters, measurement of the specific heat.

The experiences in the Laboratory will cover:

• Basic measurements of physical quantities

• Free body fall

• Rotations and torsion pendulum

• One-dimensional harmonic motion

• Bernoulli and Poisson distributions

• The adiabatic calorimeter

## Bibliography

• J.R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, University Science Books.

• Additional material provided by the lecturer: all the slides used by the teacher are abvailable on Elly website.

## Teaching methods

Oral lesson and laboratory, alternated, with strong prevalence of laboratory activities.

Oral lectures will be held in presence, respecting the safety standards, with the possibility to visualize their registration (for a defined time) by remote in asynchronous way (on Elly or Stream).

Oral lectures will be used to furnish the physical basis of the experiments and to explain the theory of errors and measurements. The main part of the module is composed by six laboratory experiences, composed buy 4 hours each. The laboratory activities will be carried on in small groups (respecting the safety procedures) to encourage the personal active participation.

PC will be used for the data acquisition and processing.

All the presentations will be available from the beginning on the Elly platform. The slides are the main didactical support for the experimental part. The main instrument for the study of the theory of measurement and errors is the suggested book. The subscription to the course on Elly is mandatory to obtain access to files, forum, to upload the reports , to check the corrections and votes.

## Assessment methods and criteria

In Itinere evaluations plus final oral examination. The laboratory work is accounted for by written reports, one for each laboratory experiment. The corrected reports and their evaluation will be uploaded on Elly. In case of not positive evaluation during the course, a laboratory experience or suplementary written or oral tests could be required. The final vote of the module (0-30) is a combination of the in itinere evaluation (written reports, nearly 20-25%) and the oral examination (75-80%).

The final vote of the course will be the average of the evaluations of the two modules.

## Other information

The course is split up into two periods: 6 CFU in the first semester and 6 CFU in the second semester. There is a single final exam at the end of the second semester.

The active participation of the student in the laboratory experiences is an essential part of the course, as well as of the evaluation path.

In the presence of special conditions (for example, in the case of working students) the opportunity to carry out personalized courses can be assessed.

In order to access the final evaluation, 70% of the laboratory lessons are required. In case of high level of absences, or when the participation in one or more experiments was precluded, recovery activities can be evaluated.

Office hours: Wednesday, 10.30-11.30 or upon appointment.