## Learning objectives

This 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 course the student will be able to: plan simple experiments of Physics, evaluate and treat the statistical and systematic errors of measurement. It will also have acquired a familiarity with the different methods of measurement and the ability to process and analyze statistically the results of measurements by means of suitable tools that help also their graphical representation, and summarize relations with the experiments themselves. The student will learn the basic concepts of probability theory, the key statistical distributions and their properties, the main statistical methods for data treatment.

## 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

Metrology: base and derived physical quantities, units of measurements in mechanics, measuring instruments, characteristic of measuring instruments (accuracy, precision, promptness, dynamic range), graph representations of data.

Uncertainty in measurements: systematic and casual errors, uncertainty propagation in indirect measurements, statistical methods in data treatment, random variables, frequency distributions, bad data treatment, weighted mean, gaussian distribution.

Correlation coefficient, best fitting and regression, chi-squared tests.

Basics of theory of probability: statistics and probability, stochastic variables, discrete and continue events, events and sample spaces, dependent and independent events, conditional probability, probability distributions (Normal, Binomial, Poisson, etc.), estimators and their properties, distribution functions and probability density functions, law of very large numbers, central limit theorem.

The laboratory experiments will be defined with reference to the topics treated in the Course of Physics 1 and will cover the following subjects:

- free body fall

- motion of rigid bodies

- motion of pendulum

- harmonic oscillations

- fluid mechanics

- waves in continuum media

- calorimetry

## 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

• Composition of Forces

• One-dimensional harmonic motion

• Motion of simple pendulum

• Bernoulli and Poisson distributions

• The adiabatic calorimeter

Module II

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

2. 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.

3. Discrete probability distributions: discrete uniform distribution; binomial distribution: moments, recurrence relations; Poisson distribution: moments.

4. 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.

5. 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.

6. 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.

7. 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

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

- Additional material provided by the lecturer.

## Teaching methods

Oral lesson and laboratory. Lectures, laboratory exercises, lectures on computer (software facilities, scientific computing, acquisition and treatment of data, experiment simulations).

## Assessment methods and criteria

In Itinere evaluations. Joined oral and written exam.

The laboratory work is accounted for by written reports, one for each weekly laboratory experiment. During the course, some written exercises concerning the theory and the laboratory experiences are proposed. At the end of the course an oral and written examination and, in case of not positive evaluation during the course, a laboratory experience is required.

## 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.

## 2030 agenda goals for sustainable development

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