## Learning objectives

This course aims to bring students to a level of knowledge in measurement methodologies that enable them to manage independently simple laboratory experiments to determine mechanical and calorimetric quantities. Also aims to provide students with a basic knowledge of the theory of errors with elements of probability theory and stochastic variables.

Knowledge and understanding

The student will learn the basic concepts of probability theory, the key statistical distributions and their properties, the main statistical methods for data treatment. The student will be able to discuss basic Physics topics in order to devise and carry out the experimental verification.

Applying knowledge and understanding

At the end of the course the student will be able to: plan simple experiments of Physics, evaluate and treat the statistical and systematic errors of a measurement. It will also have acquired a familiarity with the different methods of measurement and the ability to process and analyze statistically the results by means of suitable tools that help also their graphical representation and summarize relations within the experiments themselves.

Making judgements

Through working in the lab, discussions with the teacher and the team group, students will be led to analyze and evaluate the scientific method, also throughout observational process, experimental proof, and critical revision of results. Usage of the hypothetico-deductive model approach will be implemented, together with the ability, in devising new experiments, to differentiate between essential and marginal aspects.

Communication skills

After every activity in the lab students will produce a written work to develop abilities in description, presentation and discussion of their results. Continuous interaction with teacher and other students will develop also the oral communication skill.

Learning skills

Lab activity, necessity to provide new or different solutions to a problem, will develop both analytical and creative skills in problem-solving, thus widening the students way of thinking.

## Prerequisites

The student has to have attended the first module of the teaching.

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. Every basic concept in Physics, however, will be discussed along with the introduction to each experiment in the lab.

## Course unit content

The course is divided into two parts: lectures and experiments in the laboratory. Lecture will be mainly devoted to discuss the fundamentals of theory of errors and statistics in order to be able to carry on an adequate analysis of experimental data.

In brief the lectures will be devoted to the following items.

Basics of theory of probability.

Distribution functions for discrete and continuous random variables.

Estimators and their properties.

Statistical hypothesis testing

Laboratory experiments

- motion of rigid bodies

- motion of pendulum

- fluid mechanics

- waves in continuous media

- calorimetry and phase transitions

## Full programme

Unit II

1. Basics of theory of probability: statistics and probability. Short account on the axiomatic theory of probability: axioms of Kolmogorov.

2. Fundamental theorems of the theory of probability: addition and multiplication of events; complement of an event; dependent and independent events; conditional probability. Addition and multiplication rules for independent and dependent events; total probability theorem; Bayes’ formula. Repeated trials: Bernoulli trials, binomial law. Short account on the deduction of the theorems in the frame of axiomatic theory of probability.

3. Probability distributions: distribution laws, cumulative distribution functions and probability density; estimators and their properties: mean, median, mode; moments of a distribution, asymmetry and kurtosis. Chebishev inequality.

4. Discrete probability distributions: discrete uniform distribution; binomial distribution: moments, recurrence relations; Poisson distribution: moments. Radioactive decays.

5. Continue probability distributions: continue uniform distribution; Gauss distribution; standardized gaussian distribution; moments; gaussian approximation of binomial and Poisson distributions. Central limit theorem. Chi-squared distribution. Cauchy distribution.

6. Gaussian distribution: maximum likelihood criterion: mean as the best estimate, standard deviation, standard deviation of the mean, weighted average. Demonstrations of relations for error propagation: basic operations, sum of squared errors, general formula.

7. Applications to data treatment: least squares fitting and regression, linear fitting, weighted least squares fitting; non-linear fitting. Multiple stochastic variables, marginal density, stochastic independence, covariance; covariance and error propagation. Correlation: linear correlation coefficient.

8. Applications to data treatment: consistency tests: significance level, chi-squared test; consistency of a distribution.

The laboratory experiments will cover the following subjects:

• motion of rigid bodies

• motion of pendulum

• torsional oscillations

• damped and forced oscillatory motion

• fluid mechanics

• waves in continuum media

• calorimetry and phase transitions

## Bibliography

1. - J.R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, University Science Books, Third edition 2022.

2. M. Loreti, Teoria degli errori e fondamenti di statistica, http://wwwcdf.pd.infn.it/labo/INDEX.html (2005).

3. - Additional material provided by the lecturer: notes of the lectures will be available to students on Elly platform.

## Teaching methods

The didactic activities are divided into classroom lessons and practical laboratory activity. The module is 6 CFU (credits). Classroom lessons are 2 credits that correspond to a total of 14 hours of classroom activity. The practical laboratory activity is 4 credits, which corresponds to a total of 48 hours of laboratory activity. The slides used to support classroom lessons will be uploaded weekly on the Elly platform. To download the slides, you need to enroll in the online course. Slides are considered an integral part of teaching material. Part of the frontal lessons will be dedicated to the detailed description of the laboratory experiments. In the laboratory will also be presented both the instrumentation to be used and data acquisition and analysis programs.

The practical laboratory activity might be partially modified to adhere the safety lineguides that will be available at that time in case of SARS-Cov2 pandemic perduring.

## Assessment methods and criteria

In Itinere evaluations. Oral examination.

The laboratory work is accounted for by written reports, one for each laboratory experiment. All the written reports will receive at the end of the semester a final mark (0-30). At the end of the course an oral examination (0-30) and, in case of not positive evaluation of the written reports during the course, a laboratory experience might be required. The final graded will be a weighted average between the written reports (40%) and the oral examination (60%). In case of a partial oral examination at the end of the first semester a partial examination also at the end of the second semester will occur.

## 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.

At least 70% of all the laboratory activities have to be fulfilled. In case of particular needs (i.e., as an esample, working students) the design of personalized solutions will be evaluated.