POSITION%20AND%20COORDINATES

1
POSITION AND COORDINATES

• to specify a position, need
• reference point (origin) O,
• distance from origin
• direction from origin (to
  define direction, need reference direction(s)
• position along a line
• position specified by one (signed) number
• position in a plane
• position of point P specified by length of
  vector OP (distance)and angle of OP with
  respect to reference direction,
• or by two numbers x,y
• position in 3-dimensional space
• need a third number (e.g. height above the x-y
  plane)
• coordinates
• set of numbers to describe position of a point

2
VECTORS AND SCALARS

• physical quantities can be scalars, vectors,
• tensors, ......
• scalar
• quantity for whose specification one number is
  sufficient
• examples mass, charge, energy, temperature,
  volume, density
• vector
• quantity for whose specification one needs
• magnitude (one number)
• direction (number of numbers depends on
  dimension)
• numbers specifying vector components of the
  vector in suitably chosen coordinate system
• e.g. components of the position vector numbers
  specifying the position
• examples position vector,
  velocity, acceleration, momentum, force, electric
  field,..
• magnitude length of vector e.g.
• distance from reference point magnitude of
  position vector,
• speed magnitude of velocity.

3
velocity

• velocity (change in position)/(time interval)
• average velocity velocity evaluated over finite
  (possibly long) time interval vav ?x/?t,
  ?x total distance travelled during
  time interval ?t (including speeding up, slowing
  down, stops,...)
• instantaneous velocity velocity measured over
  very short time interval
• ideally, ?t 0, i.e. time interval of zero
  length v limit of (?x/?t) for ?t? 0
  ?t? 0 is limit of ?t becoming
  infinitesimally small, ?t approaches zero,
  ?t goes to zero
• note that velocity is really a vector quantity
  (have considered motion in only one
  dimension)
• difference quotient ?x/?t difference
  quotient of position with
  respect to time
• difference quotient ratio of two differences
• limit for ?t? 0 limit of (?x/?t) for ?t?
  0 dt/dx differential quotient, also
  called derivative of x with respect to t
• differential calculus branch of mathematics,
  about how to calculate differential quotients.
• angular velocity ? (change in angle)/(time
  interval)
• ? 2 ? f (f frequency of rotation)

4
ACCELERATION

• acceleration rate of change of velocity
• a (change in velocity)/time interval
• average acceleration aav ?v/?t , ?v
  change in velocity ?t duration of time
  interval for this change
• instantaneous acceleration limit of
  average acceleration for infinitesimally short
  time interval , a dv/dt
• acceleration, like velocity, is really a vector
  quantity
• change of velocity without change of speed
• if only direction changes, with speed staying the
  same
• e.g. circular motion
• if a 0 no acceleration, ? velocity
  constant ? uniform motion motion in
  straight line with constant speed
• angular acceleration rate of change of
  angular velocity