Summer School 2007B. Rossetto1 French Summer School Phnom Penh 2007 Mechanics I ROSSETTO Bruno Institut Universitaire de Technologie Université du Sud-Toulon-Var (France) tél site:

Summer School 2007B. Rossetto2 Mechanics I  Summary Chap. 1 – Coordinates Chap. 2 – Vectors Chap. 3 – Differential operators Chap. 4 – Forces. Equilibrium Chap. 5 – Kinematics. Particle motion Chap. 6 – Relative motion Chap. 7 – System of particles Chap. 8 – Rigid body motion

Summer School 2007B. Rossetto3 Mechanics I  References M. Alonso and E. J. Finn, Fundamental University Physics, vol. 1 Mechanics, Addison Wesley (1969) C. Kittel, W. D. Knight, M. A. Ruderman, The Berkeley Course on Physics, vol. 1 Mechanics, Mc Graw Hill, (1965) R. W. Feynmann, M. Leighton and M. Sands, The Feynmann Lectures on Physics, vol 1, Mainly Mechanics, Radiation and Heat, Addison Wesley, early 1960s)

Summer School 2007B. Rossetto4 1. Coordinates  Cartesian x y 0 2-dim. 1 - Origin System of orthogonal axis (0xy) 3 - Unit vectors and x y 0 z 3-dim. Orientation of the three-dimensional system of coordinates: - screw rule - right hand rule

Summer School 2007B. Rossetto5 1. Coordinates  Orientation rules x y z y z x

Summer School 2007B. Rossetto6 1. Coordinates  Orientation rules x y z y x z

Summer School 2007B. Rossetto7 1. Coordinates x y 0  Polar (2-dim.)   Cylindrical (3-dim.) P(r,) x 0 z  P(r,,z) P(r,) and For both: and  3-dim.:

Summer School 2007B. Rossetto8 1. Coordinates  Transformations  r x y z 0 r z 1 – From polar to cartesian x = r cos y = r sin z = z 1 – From cartesian to polar z = z

Summer School 2007B. Rossetto9 1. Coordinates  Spherical   r r sin  x y z 0 z 2 - Transformations 1 - Definitions

Summer School 2007B. Rossetto10 1. Coordinates System of coordinates Differential line elements along the coordinate axis of the system Cartesian (x, y, z)dx, dy, dz Cylindrical (r, , z)dr, r d, dz (cf applications 2, 4) Spherical (r, , )dr, r sind, r d (cf applications 5, 6) Definition of radian (for the disk : = 2 radians) r 0 r  A B From this definition:

Summer School 2007B. Rossetto11 1. Applications (1) 1 – Triangle area from the equation b pb = h x 0 h r x 0 -r 2 – Surface of a disk from the equation If b is the basis and h the height: - Find the equation of the line OA - Use a property of integrals A - Find the equation of the circle - Use a property of integrals

Summer School 2007B. Rossetto12 1. Applications (1) 1 – Triangle area from the equation b f(x) = px pb = h x 0 h r x 0 -r 2 – Surface of a disk from the equation If b is the basis and h the height: Equation: f(x)=px

Summer School 2007B. Rossetto13 1. Applications (1) 1 – Triangle area from the equation b f(x) = px pb = h x 0 h r x 0 -r 2 – Surface of a disk from the equation Let If b is the basis and h the height:

Summer School 2007B. Rossetto14 1. Applications (2) 0 r d r  r 2 - Surface of a disk using polar cordinates The contribution to the area of the sector having r as length and  as angle is the aerea of the triangle having r as basis and rd as height: r 0 r dd A B 1 - Length of a circonference Contribution of the angle d to the length: Total length: sum of contributions: Total area : A= dA=

Summer School 2007B. Rossetto15 1. Applications (2) 0 r d r  r 2 - Area of a disk using polar cordinates The contribution to the area of the sector having r as length and  as angle is the aerea of the triangle having r as basis and rd as height: Total aerea : r 0 r dd A B 1 - Length of a circonference Contribution of the angle d to the length: Total length :

Summer School 2007B. Rossetto16 1. Applications (3) 1 – Surface of a triangle (base b, height: h) b f(x)=px pb=h x 0 h 2 – Surface of the ellipse 0 a b dA= A= Contribution of the infinitesimal surface dy.dx : dA = Equation:

Summer School 2007B. Rossetto17 1. Applications (3) 1 – Surface of a triangle (base b, height: h) b f(x)=px pb=h x 0 h 2 – Surface of the ellipse 0 a b Area: Contribution of the infinitesimal surface dy.dx : dA = dy.dx

Summer School 2007B. Rossetto18 1. Applications (4)  x y z  Cylinder area 0 r x y z 0 dz h rd 1 - Double integral: contribution of the element of length r d height dz: rddz 2 - Simple integral: contribution of the element of length 2r height: dz: 2rdz dz dA= A= dA= A=

Summer School 2007B. Rossetto19 1. Applications (4)  x y z  Cylinder area 0 r x y z 0 dz h rd 1 - Double integral: contribution of the element of length r d height dz: dA=rddz 2 - Simple integral: contribution of the element of length 2r height: dz: dA=2rdz dz

Summer School 2007B. Rossetto20 1. Applications (5)   r sin  x y z rd r length: 2 r sin, width: r d Total area: A=  Sphere area using symetries (simple integral): contribution of the element:  Sphere area using double integral Contribution of the element lenght : r sin d, width: r d r sin  0 0  r rd rsind Area : A = sum of contributions dA=

Summer School 2007B. Rossetto21 1. Applications (5)   r sin  x y z rd r length: 2 r sin, width: r d dA = 2 r 2 sin d Total area:  Sphere area using symetries (simple integral): contribution of the element:  Sphere area using double integral Contribution of the element lenght : r sin d, width: r d r sin  0 0  r rd rsind dA = r 2 sin d d

Summer School 2007B. Rossetto22 1. Applications (6)  r r sin  x y z r d r sin d r  Sphere volume (or mass if homogeneous) Contribution of the element length : r sin d weidth : r d height : dr r sin  0 0  

Summer School 2007B. Rossetto23 1. Applications (6)  r r sin  x y z r d r sin d r  Sphere volume (or mass if homogeneous) Contribution of the element length : r sin d weidth : r d height : dr r sin  0 0  