EMEM:359  Dynamics Session:19983

Test # 1: Ch. 12 , 13  - 1 Formula sheet, closed book
Friday, April 2, 1999

Rectlilinear motion:  [also 1 D motion] - vector representation is an overkill as only one direction is involved
position:             r(t)
vector:                v(t): = dr/dt                       r₂-r₁ = integral{v(t)dt} between t₁,t₂
acceleration:     a(t) = dv/dt = d²r/dt²          v₂-v₁ = integral{a(t)dt} between t₁,t₂

also               a(r)dr = vdv      0.5( v₂² -v₁²  )= integral{a(r)dr} between r₁,r₂

For convenience - dependence on t is not explicitly transcribed

Position vector:       r(t) [ 2D];    R(t) [3D] 
Velocity vector       v(t) = dR(t)/dt; 
Acceleration vector   a(t) = dv(t)/dt = d²R(t)/dt²

Cartesian/Rectangular Coordinate System:
Coordinates : x, y, z
Unit vectors :   i, j, k

R = x i  + y j  + z k; 
v  = v_x i  +  v_y j  + v_z  k   =      dx/dt i  +  dy/dt j  +  dz/dt k 
a  = a_x i  +  a_y j  + a_z  k   =      dv_x  /dt i  +  dv_y /dt j  +  dv_z  /dt k 
                                       =     d(dx/dt ) /dt i  +  d(dy/dt) /dt j  +  d(dz/dt ) /dt k 

The cartesian coordinate system can be regarded as a vector sum of 
three rectilinear motions in x, y, z, diections

Normal - Tangent - Binormal system
Coordinates:s
Unitvectors: e_t , e_n , e _b

Normal Tangent Coordinate System [ Click Button to Open in new window - uses Javascript ] 

Spherical Coordinate System [ Click Button to Open in new window - uses Javascript ] 
 

Relative Motion[ Click Button to Open in new window - uses Javascript ] 
 

Newton's Law : F = m a {independent of coordinate system}

Newton's Law of Gravitation: 

                               F = m₁m₂ / R²   (The forces are directed along their centers)

Rectangular System: 
 

     Conservative Forces:  Forces whose work is INDEPENDENT of path but depends on position;
           Examples ; Weight and Elastic Spring force

     Each conservative force can be associated with a potential function (V) which depends on position 
           F  = - grad{ V }

     V_g= potential function due to weight  =  mgy ; m = mass of particle,  g = gravitational acceleration
                                                                              y = distance above datum  (reference for zero potential)
     Work done when the particle moves from A to B  = V_gB - V_gA

      V_e = potential function/energy due to elastic spring  = 0.5 k x² ; k = spring constant, x = change from unstretched length
     Work done when the partices moves from A to B and spring is involved = V_eB - V_eA

Plane Translation -  angular acceleration= 0

Plane Rotation - linear acceleration = 0