Learning objectives
This course aims at bringing students to a level of knowledge in measurement methodologies that gives them autonomy in dealing with simple laboratory experiments to determine mechanical anc clorimetric quantities. Also aims at providing students with a basic knowledge of the error analysis with elements of probability and and stochastic variables theory.
At the end of the course the student will be able to: plan simple Physics experiments, evaluate and treat the statistical and systematic errors of measurement. they will also have to acquire 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 main 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 an object, 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
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
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.
Teaching methods
Oral lesson and laboratory activity. Lectures, laboratory activity, 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 examinations.
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 will be 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 will be required.
Other information
The course is split up into two parts: 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: Monday, 14.30-16.30 or upon appointment.
