081022

 

A vector of magnitude R, lying at angle theta as measured in the counterclockwise direction from the positive x axis, has x and y components R_x and R_y, corresponding to its projections onto the x and y axes, which are equal to
  • R_x = R cos(theta) and
  • R_y = R sin(theta).

If a number of vectors are added, the x component of the sum is equal to the sum of the x components, and a similar statement applies to the y components.

If a vector has components R_x and R_y, then its angle as measured counterclockwise from the positive x axis is

  • theta = arcTan(R_y / R_x), plus 180 degrees if R_x is negative

and its magnitude is

  • R = sqrt( R_x^2 + R_y^2).

The vector R from point (x1, y1) to point (x2, y2) on a Cartesian coordinate system has components

  • R_x = (x2 - x1) and
  • R_y = (y2 - y1).

Find the x and y components, then the magnitude and angle of the vector from (2, 4) to (5, 9).

Find the x and y components of the vector whose standard angle is 140 degrees and whose magnitude is 7.

Find the x and y components of the sum of the two vectors found in the preceding two questions.  Then find the magnitude and angle of the sum.

Find the magnitude and angle of each of the three length vectors in the rubber band experiment.  Find the sum of the length vectors.

Find the magnitude and angle of each of the three force vectors in the rubber band experiment.  Find the sum of the force vectors.

Forces of 5 N at standard angle 60 degrees and 4 N at standard angle 140 deg are exerted on a paperclip by two rubber bands.  A third rubber band balances these two forces so that the paperclip remains stationary. 

  • What are the magnitude and angle of this third rubber band's force? 
  • If the rubber band exerts a force of .8 Newtons for every centimeter of its length in excess of 7.5 cm, what is its length and what are the x and y components of its length vector?

A jet plane having mass 50,000 kg is traveling at 900 km / hr at an altitude of 37,000 feet as it begins its descent into Atlanta.  13 minutes later it is traveling at 937 km/hr and its altitude is 30,000 feet. 

  • By how much have the kinetic and potential energies of the plane changed? 
  • By how much has its total mechanical energy (sum of potential and kinetic) changed? 
  • How much work was done on the jet by nonconservative forces during this interval? 
  • How much work was done by the jet against nonconservative forces? 
  • How much work was done on it by gravity? 
  • How much work did it do against gravity?  How much work was done on the jet by the net force acting on it?