Download Traveling Washer in Two Dimensions

Traveling Washer in Two Dimensions
Materials:
Small Washer or Coin
Centimeter Ruler
Centimeter Graph Paper (there are free programs that will do this
on your computer – a Google search for ‘graph paper’ finds one)
Small Magnetic Compass
Tape (Masking or Scotch)
Protractor
1. On the piece of centimeter graph paper, draw a vertical line in the center of the paper to
represent the Y-axis, and draw a horizontal line in the center of the paper to represent the Xaxis.
By convention the top of the paper will represent north (N), the right side east (E), the bottom
south (S), and the left side west (W)
2. Place the center of a washer (or coin) at the southwest corner of the graph paper. Align the
center of the hole in the washer with the intersection of a horizontal and a vertical line. Draw
a circle around the inside of the washer. Mark the center of this circle with the letter “I” for
the Initial position of the washer.
3. Move the center of the washer in the manners listed below. After each move, draw a circle
around the washer.
a.
b.
c.
d.
e.
f.
g.
2.0 centimeters North
5.0 centimeters East
5.0 centimeters North
3.0 centimeters West
4.0 centimeters North
6.0 centimeters East
3.0 centimeters South ….. STOP ….. Label this circle “F” for the Final position of
the washer.
4. Find the total distance the washer traveled. __________________________________
_____________________________________________________________________
5. Using a centimeter ruler, draw a line from the “initial” position of the washer to the “final”
position of the object. Measure the length of this line using the centimeter ruler. What is this
measured distance called?
___________________________________________________________________________
6. Which is longer, the total distance traveled or the line from the initial position to the final
position?
___________________________________________________________________________
Jane and Jim Nelson PTRA Workshop Leader’s Manual Teaching about Motion, ©2001 AAPT, Nelson
7. In what direction is the “final” position of the washer in comparison to the “initial” position?
Circle one of the following:
a. Directly North
e. Northeast
b. Directly South
f. Northwest
c. Directly East
g. Southeast
d. Directly West
h. Southwest
7. Write three different descriptions or instructions for how to get from the “initial” position of
the washer to the “final” position of the washer.
DESCRIPTION 1: ___________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DESCRIPTION II: ___________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
DESCRIPTION III: __________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
8. Compare your instructions with instructions written by other participants. What are the
similarities and differences among the various instructions you read? What are the
characteristics of a good set of instructions?
__________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Jane and Jim Nelson PTRA Workshop Leader’s Manual Teaching about Motion, ©2001 AAPT, Nelson
Worksheet
Vectors are physical quantities that have both a magnitude and a
direction. The rules for vector addition apply to all vector quantities i.e:
53o
displacement, velocity, acceleration, force, momentum, and impulse. When
adding vectors, the order of the vectors does not matter. For example, if you
walk 4 m north and then 3 m west you end up in the same location as you would
if you walk 3 m west and then 4 m north. The vector sum (called resultant) will
be a straight-line vector that begins at the tail (the start) of the initial vector and ends at the tip
(head or arrow) of the final vector. If the vectors are at right angles, use the Pythagorean theorem
to determine the magnitude, 3  4
 5 , and trigonometric functions to determine the
angle, arc tan  = magnitude of the vertical component/magnitude of the horizontal component
or arc tan  = 4 m/3 m, or  = 53o. Vector notation requires both the magnitude of 5 m as well as
the direction, 53o NW, north of west. You could have specified the direction as 37o WN west of
north, although this is not as common a form as measuring from the horizontal axis. Notice that
the vector has the same direction with respect to the rectangular coordinate system. It is often
easier to measure angles from a horizontal reference line. This aids in graphical addition of
vectors as well as making vector diagrams and identifying components of vectors. Note that a
designation such as NW or SE with no numberical angle shows means exactly 45o.
2
2
The solutions to the problems below can be found by drawing them on graph paper and using a
ruler and protractor to find the displacements or by calculating resultants using trig.
1. Move your pencil 5.0 centimeters west and then 8.0 centimeters north of the origin. Mark
the final point.
a. How many centimeters did your pencil travel all together? (What was the distance
traveled by your pencil?)
__________________________________________________________________
b. How many centimeters from the origin is your pencil now located? (What is the
displacement of your pencil? Remember to include direction.)
__________________________________________________________________
2. Find the displacement of a point that is 6.0 centimeters east and 8.0 centimeters north of
the origin.
______________________________________________________________________
3. Find the displacement of a point that is 4.0 centimeters west and 8 centimeters south of
the origin
_______________________________________________________________________
Jane and Jim Nelson PTRA Workshop Leader’s Manual Teaching about Motion, ©2001 AAPT, Nelson
Traveling Washer in Two Dimensions
Teacher’s Notes
Idea:
The movement of an object can be described in terms of initial position, final position,
distance traveled, and displacement. Position can be specified by rectangular coordinates
or with polar coordinates (if your prefer to use them).
Management tips:
Be sure that the final position of the washer will be on the paper. This activity takes about
30 minutes.
Measured distances may vary depending on how the participants measure (e.g., from
center to center of the washer or from outside edge to inside edge). You may choose to
set the method of measurement OR simply discuss the differences in distances and
possible causes.
Responses to some questions:
The total distance traveled is 28 centimeters; however, the ‘final’ position is 8.0
centimeters north and 8.0 centimeters east of the “initial’ position. The displacement
(i.e., straight line from “initial’ to ‘final’) is 11.3 centimeters.
The final position is to the northeast of the starting position.
Written directions will vary. There are three simple directions. The first is: go 8.0
centimeters east, and then 8.0 centimeters north. The second is: go 8.0 centimeters north,
and then 8.0 centimeters east.
Directions of this type are like plotting points on a graph, and makes use of rectangular
coordinates.
Directions of another type are: go 11.3 centimeters at an angle of 45 degrees east of
north. Directions of this latter type are commonly used in navigation, and make use of
ranging or polar coordinates. The angle is often called the azimuth angle.
Points to Emphasize in summary discussion:
Scientists distinguish between the length of the path traveled by an object and the length
of the straight line drawn from the starting position to the final position. The length of
the path traveled is called DISTANCE TRAVELED, while the length of the line from the
starting position to the final position (including the Direction) is called the
DISPLACEMENT.
Jane and Jim Nelson PTRA Workshop Leader’s Manual Teaching about Motion, ©2001 AAPT, Nelson
Background Information
There are several methods for locating a point in space.
The rectangular coordinate system uses North-South
and West-East coordinates. On a piece of graph paper
the rectangular coordinates of a location (i.e., point) is
generally expressed as (x,y), which are referred to as
the x-coordinate, and the y-coordinate. On the surface
of the Earth this is commonly referred to as latitude
and longitude. The origin for latitude is the North Pole West
with angles of latitude measured going south and the
South Pole with angles of latitude going north. The
equator is 90 degrees latitude in both cases. The origin
for longitude is the prime meridian, which is a line
from the North Pole to the South Pole through
Greenwich, England. See diagram #1. A point is
specified by P(x,y).
North
East
South
[Diagram #1 Rectangular Coordinate System]
Responses to Worksheet questions:
1. a) 13 cm
b) 9.4 cm 60o NW
2. 10 cm 53o NE
3. 8.9 cm 63o SW
Jane and Jim Nelson PTRA Workshop Leader’s Manual Teaching about Motion, ©2001 AAPT, Nelson