## Learning objectives

Acquirement of basic measurement techniques and methodologies for experiments in Mechanics and Calorimetry, including software for data acquisition and analysis. Theory of errors and basics of probability theory. Acquirement of measurement techniques and methodologies concerning the proposed experiments.

## 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); measurement errors and their representation, systematic and random errors, confidence intervals.

Uncertainty in physical measurements: statistical methods in data treatment, uncertainty propagation in indirect measurements, statistical analysis of random errors, frequency distributions, gaussian distribution, bad data treatment, weighted mean. Least-squares fitting and regression; covariance and correlation, consistency 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, estimators and their properties, distribution functions and probability density functions. Analysis of specific distributions (uniform, gaussian, Binomial, Poisson, Cauchy, ...), law of large numbers, central limit theorem.

Short account on calorimetry: definition of temperature, temperature measurement techniques, thermocouple, specific heat and heat capacity. Heat transfer mechanisms, calorimeters, measurement of specific heat.

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. Metrology: base and derived physical quantities, units of measurements in mechanics, measuring instruments, characteristic of measuring instruments (accuracy, precision, promptness, dynamic range); measurement errors and their representation, order of magnitude and significant digits.

2. Uncertainty in physical measurements: error propagation in indirect measurements (basic operations, sum of squared errors, one- and multiple-variables function); error as differential. Measurement errors and their representation: confidence interval, significant digits, consistency and discrepancy between data, test of physical laws.

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

4. Uncertainty in physical measurements: frequency and probability, limit distribution, probability density; normalization, mean and standard deviation for a distribution. Gaussian distribution: confidence and standard deviation, error function; data comparison. The mean as the best estimate. Population distributions.

5. Uncertainty in physical measurements: bad data treatment, weighted mean (Chauvenet criterion); Short account on Least squares fitting and regression.

6. Basics of theory of probability: statistics and probability, stochastic variables, discrete and continue events; favorable cases and total cases; definitions of probability.

7. Combinatorics: counting assignments; selections with and without replacement; permutations; combinations; distributions of objects to bins; games.

8. Short account on calorimetry: definition of temperature, temperature measurement techniques, thermocouple, specific heat and heat capacity. Heat transfer mechanisms, calorimeters, measurement of specific heat.

The laboratory experiments will cover the following subjects:

• basic measurement of physical quantities

• free body fall

• force composition

• one-dimensional harmonic motion

• simple pendulum

• Bernoulli and Poisson distributions

• Regnault calorimeter

Module 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, Introduzione all'Analisi degli Errori, Ed. Zanichelli, Bologna, 2° ed., 2000.

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

3. Subsidiary material provided by the teacher.

For further insights: R. Scozzafava, “Primi passi in probabilità e statistica”, Ed. Zanichelli, Bologna, 2° ed., 2000.

## 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.

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