28626 - General Physics T-A

Learning outcomes

At the end of the course the student has assimilated and is able to apply the knowledge on the basic concepts of the General Physics in the language of the Mathematical Analysis, of the Integral and Vector Calculus; he has assimilated and is able to apply the technical-scientific methodology needed in order to face in quantitative terms the physics problems.

Course contents

Rudiments of Vector Calculus. Scalar, vector and tensor physical quantities. Vectors, versors (unit vectors) and vector components. Vector sum and its properties. Rotations and their non-vector nature. Vector difference. Triangle inequality. Scalar multiplication of a vector and its properties. Dot product between two vectors and its properties. Algebraic expressions of the vector components with respect to a given oriented direction. Square of a vector. Algebraic expression of the vector norm. Vector product between two vectors and its properties. Scalar triple product. Cartesian representation of the vectors. Relation between the versors of an orthonormal cartesian triad. Vector operations in the cartesian representation. Derivative of a point and of a vector. Integral of a vector. The nabla symbol. Gradient, divergence and curl. Review of integral types used in physics: simple, double and triple integral of a scalar or vector function; line integrals of a scalar and vector field; surface integral of a scalar and vector fields.

Kinematics. Point particle. Reference frames and Cartesian coordinate triads. The principle of special relativity. Time intervals and their measurement. Short history of the time unit. Short history of the length unit. Intrinsic (or local) description and Cartesian description of the motion: trajectory and distance along the path as a function of time versus parametric equations of motion. Average and instantaneous speed. Limits to the concept of instantaneous speed: the instantaneous speed of a free electron. Average and instantaneous velocity. Cylindrical coordinate system. Intrinsic, Cartesian and cylindrical representation of velocity. Areal velocity. Average and instantaneous acceleration. Intrinsic representation of acceleration. Osculating plane, osculating circle and radius of curvature of the curve at a point. Cartesian and cylindrical representation of acceleration. Constraints and degrees of freedom. Kinematics of the rigid bodies. Poisson formulae. Fundamental formula of the kinematics of the rigid bodies. Translational motion, rotative motion. Pure rolling. Angular velocity.

Galilean Transformations. Change of reference frames. Galileo's transformations. Transformation of the position vector and of the velocity. Drag velocity. Transformation of the acceleration. Drag acceleration and Coriolis acceleration.

Applied Vectors. Polar and axial moment of an applied vector. Resultant, resultant moment and axial resultant moment of an applied vector system. Centre of parallel vectors. Equivalent systems of applied vectors. Couple of applied vectors. Reduction of an applied vector system to a vector and a couple.

Statics. Static definition of a force. Dynamometer. Hooke's law. Equilibrium of a system of forces. The vector nature of a force. Internal and external forces. Cardinal equations of statics (equilibrium equations). Centre of gravity (or barycenter) and its properties. Active and constraint forces. Friction forces. Sliding friction and rolling friction. Static and kinetic sliding friction. Limiting friction.

Point Particle Dynamics. Motion in pre-Galilean natural philosophy. Galileo's experiments on inertia. The Newton's first law of motion (law of inertia) in classical and modern formulation. Inertial reference frames. The principle of special relativity. The reference frame of the fixed stars and the reference frame of the Earth surface. Absolute space and Mach's principle. The Newton's second law of motion. Mass and density. Units of measurement and dimensional analysis. Mass and weight. Momentum and impulse. Impulse-momentum theorem. Dynamical definition of a force. Kepler's laws and Newton's law of universal gravitation. Cavendish experiment. Inertial mass and gravitational mass. Forces depending on position, velocity and time. Implicit and explicit time dependence. Fundamental particle dynamics problem.

Remarkable motions. Uniformly accelerated rectilinear motion. Free fall in the gravitational field of a body in vacuum. Motion of a projectile in vacuum. Range and highest trajectory point. Motion of a body in a viscous fluid. Reynolds number. Laminar flow and Stokes' law. Steady and periodic turbulent flow. Linear drag and quadratic drag. Drag coefficient. Motion of a sphere with linear drag. Free fall in the gravitational field of a sphere with linear drag. Terminal velocity. Motion of a sphere with quadratic drag. Free fall in the gravitational field of a sphere with quadratic drag. Motion of a sphere thrown upward in the gravitational field with quadratic drag. Simple harmonic oscillator. Simple harmonic motion: period, frequency and angular frequency. Simple pendulum: solution of the motion equation for small oscillation amplitudes. Damped oscillator. Over-damped, critically damped and under-damped motion. Driven oscillator. Transient state and steady state. Resonance.

Inertial forces. Reference frames in accelerated translational and rotative motion with respect to the fixed stars. Inertial forces (also known as fictitious forces, apparent forces, d'Alambert forces or pseudo-forces). Drag force and Coriolis force. Centrifugal force and Euler force. Dependency of the weight force on the latitude. Deviation towards east of the free falling bodies. Deviation of the bodies in motion on the Earth surface. Foucault pendulum. Dependency of the rotation speed of the swing plane on the latitude.

Dynamics of particle systems and of rigid bodies. Action-reaction law. Resultant and resultant moment of internal forces. Closed systems. Angular momentum. Cardinal equations of dynamics. Conservation of momentum and of angular momentum. Centre of mass. Centre of mass and centre of gravity. Properties of the centre of mass. The centre of mass theorem. Consequences of the cardinal equations of dynamics on planet motion. Angular momentum of rigid bodies: case of a rigid body rotating around a fixed axis and case of a roto-translating rigid body with the rotation axis parallel to itself. Moment of inertia. Moment of inertia of homogeneous bodies. Huygens [http://en.wikipedia.org/wiki/Christiaan_Huygens] –Steiner [http://en.wikipedia.org/wiki/Jakob_Steiner] theorem. Dynamics of systems.

Work and energy. Elementary work. Exact differentials and differential forms. Line integral of a vector field. Surface integral of a vector field. Work done by a force on a point particle moving on a curve. Examples of work calculation. Kinetic energy. The work-energy theorem. König's theorem for a system of particle points and for a rigid body. The work of the weight-force. Positional forces and conservative forces. Differential operators: gradient, divergence and curl. Physical meaning of the curl. Stoke's theorem. Properties of conservative force fields. Potential, potential energy and total mechanical energy. Conservation of mechanical energy. Collisions. Collision forces. Collinear collision between two particle points. Coefficient of restitution. Elastic and inelastic collisions. Conservation of mechanical energy, linear momentum and angular momentum in collisions processes between rigid bodies.

Readings/Bibliography

• Copy of the transparencies presented during the course, available on World Wide Web at the Virtual Learning Environment: VIRTUALE
• Question and exercises for the assessment, available on World Wide Web at the Virtual Learning Environment: VIRTUALE
• Halliday, Resnick, Krane, Physics, vol. 1, John Wiley & sons.
• Feynman, Leighton, Sands, The Feynman Lectures on Physics, vol I, Addison-Wesley.
  
• Focardi, Massa, Uguzzoni, Fisica Generale, Meccanica e Termodinamica, Casa Editrice Ambrosiana, Milano.
• C. Mencuccini e V. Silvestrini, FISICA, Meccanica e Termodinamica, Casa Editrice Ambrosiana, Milano.
• Serway e Jewett, Fisica per Scienze ed Ingegneria, Casa Editrice EdiSES.

Teaching methods

• During the frontal lessons blackboard/tablet is used and slides or multimedia are shown by means of a projector.
• Such transparencies are made available to the students before the lecture by means of World Wide Web VIRTUALE, in order to reduce the time and the work of mere transcription during the lessons and as in-depth study.
• The digital blackboards will be available on the same website within a week from the date of the lesson.
• The proposed practices demand the use of the pocket calculator.
• To communicate with students, the mailing list of University Directory Service is widely used as well as the forums provided by the VIRTUALE website.

Assessment methods

A written examination gives access to a VIVA.

Students with Specific Learning Disabilities (SLD) or temporary/permanent disabilities are advised to contact the University Office responsible in a timely manner ( https://site.unibo.it/studenti-con-disabilita-e-dsa/en ). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, and will be evaluated in light of the learning objectives of the course.

Teaching tools

Virtual Learning Environment: VIRTUALE, Blackboard, Projector, Laptop and Tablet

Office hours

See the website of Laura Fabbri