The component force cos and the component sin

• Don't let any rubber band exceed its original length by more than 30% (e.g., a rubber band with unstretched length 8 cm should not exceed 10.4 cm in length).  The rubber band most likely to exceed this length will be the one parallel to the vertical grid lines.  Keep an eye on that one as you position the other two.  It would be a good idea to check your calibration graphs to see which rubber band is capable of exerting the most force at the 'plus-30% length', and use that for the 'vertical' rubber band.

• The second rubber band will be directed toward the upper left and will be angled at least 45 degrees to the left of vertical.  This rubber band should stretched to a length that will cause it to exert a force of about 1 Newton.

Mark the positions of the ends of the rubber bands, measure their lengths and determine the force exerted by each:

1.  Using a pencil with a good point or a reasonably fine-tipped pen make an x under the end of each rubber band, with the x crossing directly under the very end of the rubber band. Locate this mark as accurately as possible.  The first figure below indicates the positions of the x's (note that the x's under the circles representing push pins will in fact be under hooks at the ends of the rubber bands). 

In the space below give the length of each rubber band and the corresponding force in Newtons, in this order and separated by a comma:

• In the first line give the information for the 'vertical' rubber band.

1.  Vertical rubber downwards after extended length is around 10 cm long plotted on the graph parallel to the vertical axis.

2. Second bar is having length around 9.97 cm sloped at 45 degree from vertical line

We will now proceed to calculate the components of the net force using three different methods.  The methods are:

• Graphical determination of force components using sketches of the vectors and projection lines.

1.  Find the point at which the lines of force meet. For each rubber band use a straightedge to draw a line along the center of the  rubber band's position.   Extend all three lines to their point of intersection.  The first figure below illustrates the lines corresponding to the above figures.

2.  Construct a y vs. x coordinate system whose origin is at the point where the three lines of force intersect, as shown in the second figure below.

5.  Determine the x and y coordinate of the point selected on each line of force.   For example in the picture the x and y coordinates of the line in the first quadrant are about 2.3 cm and 8.5 cm, so the point is (2.3, 8.5).  Note that the point (2.3, 8.5) isn't really 10 cm from the origin; when you located your points you hopefully did much better.

Report your results in the space below, x and y in centimeters in comma-delimited format, one rubber band to a line.  Report with the rubber bands in the same order you reported previously. 

3. Bar three having coordinates measured as (3,9.5)

If you use Excel you can use the ATAN function (for this angle you would type =ATAN(8.5 / 2.8) into a cell).  Excel gives angles in radians, which you can convert to degrees if you multiply by 180 / `pi (you could just type in = ATAN(8.5 / 2.8) * 180 / PI(), which will give you the angle in degrees.  Note that pi() is Excel's way of denoting `pi).  

7.  Find the angle made by each force vector with the positive x axis, as measured in the counterclockwise direction from the positive x axis.  Note that the angle you found in the preceding instruction is the angle of the line of force with the x axis. 

Report your angles in the space below, one to a line, reporting the rubber bands in the same order as before.  This will require the first three lines.

Starting in the fourth line explain briefly, in your own words, how you obtained your angles.

3. Bar Third is in first quadrant hence actual angle with + X axis = 72.4744

Your answer (start in the next line):

2.  Each vector should have its initial point at the origin, should lie along the line of force and be directed in the direction of the force exerted on the paper clip by the rubber band.  The first figure below illustrates three vectors representing forces of roughly 1.5 Newtons, .6 Newtons and 1.8 Newtons.

3.  Now draw the projection lines, sketch the component vectors, and validate the components as calculated using the sines and cosines of the angles:  From the tip of each vector sketch the projection lines back to the x axis and to the y axis. Be sure the projection lines run parallel to the grid lines and use a straightedge to locate the projection lines as accurately as possible.

Starting in the fifth line explain briefly what you numbers mean and how you obtained them.

Your answer (start in the next line):

Your answer (start in the next line):

#$&* _ total of x and of y components

At a scale of 4 cm per Newton, which can also be expressed as 1/4 Newton per cm or .25 Newton per centimeter, the resultant vector has x component -.3 cm * .25 N / cm = -.075 N and y component 1 cm * .25 N / cm = .25 N.

2.  Now sketch a vector from the initial point of the first vector to the terminal point of the last vector.  This vector is the short blue vector in the second and third figures below.  The 'blue' resultant vector originates at (0, 0) and terminates at the third point of the polygon, which we have estimated in the figure below to be at (-.3, 1). 

Give the x and y coordinates of the points on your actual graph, in cm, then the x and y coordinates of the corresponding force components, in Newtons, in the first line of the space below, in comma-delimited format.

Starting in the second line give a brief description of your polygon, including the initial and terminal points of each of the vectors in the polygon, and explain how you calculated the force components of your resultant vector using the scale of the graph.

4. Are the components of your resultant vector reasonably consistent with the results you got in the preceding activity when you added the components of the vectors?  Answer in the space  below.

Your answer (start in the next line):

#$&* _ consistency between polygon and adding components

4.  What is the magnitude of the force vector indicated by your sketch?  Use the pythagorean theorem to calculate the magnitude of the resultant force vector, as indicated by the components you gave in the preceding space .  Give your magnitude in the first line, the magnitude calculated by the Pythagorean Theorem in the second, and explain starting in the third line how you calculated it:

Add the three vectors by calculating components:

1.  For each rubber band use the magnitude and the angle of the force it exerts to determine the x and a y component of that force.

2.  What is the sum of all the x components you calculated in the preceding step? What is the sum of all your y components?

You previously obtained x and y components graphically, first using a sketch and projection lines, then using the polygon method of addition.  Compare your results with the results you obtained before.

2.  What is the resultant vector corresponding to these components? What is the magnitude of this resultant?

Report the answers to these two questions in the space  below, and explain how you know that your answers must be so:

Your answer (start in the next line):

 Max Magnitude = 2.50N

#$&* _ magnitude of percent error each way as percent of largest force vector _