POLAR COORDINATES

POLAR  COORDINATES 

Polar points are plotted using the polar coordinate plane with the line OX as initial line or polar axis, point 0 as the pole or origin and the distance of the point from O as the radius vector.
                           
The position of any point P in the plane is determined if the length of the line OP together with the angle that this line makes with OX are known, both the length and the angle being measured in a definite sense.





From the figure :
     r  = radius vector,  + if measured along the terminal side of Ө, 
                                  –  if measured in the opposite direction along the terminal side of Ө 
     Ө = polar angle
     OX = initial line or polar axis
     O = pole or origin

Polar Coordinate paper – it is where the polar points are plotted.
              ( to be illustrated on the board during lecture )

Plot the following points :

1.    ( 2, 30º )                6.  ( 3,  60º )  
2.    ( 4, 225º )              7.  ( 4,  – 315º )  
3.    ( – 6, –120º )         8.  ( – 3, –240º )
4.    ( – 4,  330º )          9.  ( 4, – 330º )
5.    ( 6, – 150º )          10. (– 6, 150º )  

Distance between two points in Polar Coordinates : by Cosine law

Conversion
A.    Polar to rectangular coordinates
         1)  x = r cos Ө
         2)  y = r sine Ө


B.    Rectangular to Polar coordinates
         1)  r = sqrt( x² + y² )
         2)  Ө = Arctan ( y/x )

Exercises :  Numbers colored red is an assignment to be passed on January 6 or 7, 2010.

1.    The point ( r , Ө ) is equidistant from ( 2 , 90º ) and (– 2 , 150º ). Express the statement into an algebraic expression.

2.    Show that the given points ( 2 , 45º ), ( sqrt( 2 ), 90º ) and ( – 2 , 135º ) are vertices of a right triangle and find its area.

3.    Convert to rectangular coordinate.
a)    ( 3 , 240º )
b)    ( 4 , 150º )
c)    (– 5 , 150º )
4.    Convert to polar coordinate
a)    ( 2 , –2 )
b)    ( – 1 , – sqrt ( 3 ) )
c)    ( – 3 , 3 ) 
5.    Find the distance between the following points.
a)    ( 3 , 240º )  and (– 5 , 150º )
b)    ( 4 , 150º ) and ( 2 , 45º )
c)    ( 2 , 90º ) and  ( – 2 , 135º )

PHYSICS E 101 Assignment ( Motion )

Physics E 101  Uniform  Motion  Assignment

1. Roy normally drives on a freeway from a city C to city A at an average speed of 90 km/hr and his trip took him 2 hours and 30 minutes. One day with heavy traffic, he slows down and drive the same distance at an average speed of 75 km/hr. How much longer does the trip take for Roy?

2. Two cyclists travel simultaneously towards each other from points 40 km apart. Cyclist A has a speed of 12 km/hr while cyclist B travels 4 km/hr slower than cyclist A. How many hours will they meet each other ? How far apart are they after 75 minutes?

3. A car is moving with an average speed of 72 km/hr. Determine (a) the total distance traveled by the car in 50 minutes, (b)the time needed to travel 90 000 m.

4. It is now 1:00 PM. At what time before 2:00 o’clock with the minute hand overtake the hour hand? At what time before 2:00 o’clock will the minute hand and the hour hand become perpendicular for the first time? At what time before 2:00 o’clock will the minute hand and the hour hand become perpendicular for the second time ?

5. Calculate your average speed in m/s and km/hr if you jog at 2.5 m/s for 10 minutes and then walk at 600 m for 400 seconds.

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HINTS for ANALYTIC GEOMETRY TRINAL EXAM

HINTS FOR TRINALS

1. Locate the point which is at a distance 4 from (7, 4) and  at a distance of sqrt(26) from (2, -1).

            Ans. (3, 4) ,  (7, 0)

2. Locate the point which is equidistant from (3, 8), (5, 2) and (-3, -4).

            Ans. (-2, 3)

3. Locate the point which is equidistant from (3, 1) and (-2, 2); also equidistant from (1, 2) and (3, 0).

            Ans.  (0, -1)

4. Show that the points (-2, 4), (3, -1) and (-1, -3) are the vertices of an isosceles triangle and find the area.

            Ans.  A = 15 sq u

5. A circle has center at (1, 2) and passes through (8, 3). Does this circle pass through (-4, -3)? Through (5, 8)? Through (0, 9)?

7. Show in two ways that the quadrilateral with vertices (0, -1), (1, 2), (-4, 7) and (-5, 4) is a parallelogram.

8. Show that the points (-1, -2), (5, 4) and (-3, 0) are vertices of a right triangle and find its area.

            Ans. 12 sq u

9. Locate the point which is equidistant from (-3, 0) and (1, 4 and at a distance 5 from (-1, 7).

            Ans. (-1, 2), (-6, 7)

10. Determine the angle from L1 along (4, 3),(6, -2) and L2 along (9, 5),(6, -2). 

            Ans. 135 degrees

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