P107. Vector Addition

Paper problems P107. Problems in Vector Addition

The protocol for this assignment is As Directed.

References: Sections 3.1-3.3 of the text; Trigonometry: Math for 2-Dimensional Physics

In this assignment, you'll use the graphical and component methods of vector addition. The problem you'll do is Chapter 3 end-of-chapter problem 20. Present your solutions as described below. Due to the nature of these exercises, you need not use the standard given, goal, etc. method.

Conventions in vector problems

The convention for giving the direction angle of a vector is to give the angle as measured counterclockwise from the +x-axis. You're expected to use this convention throughout the course.

When using a symbol to represent a vector, always make it clear that the quantity represented is a vector and not a magnitude. For example, does A represent a vector or a scalar? Our convention on a word-processed page like this is to put vectors in bold type. Hence, A is a vector and A is the vector's magnitude. On a hand-written page, the quickest method is put an arrow above the A like this: . Thus, is a vector and A is a magnitude.

Part A. Component Method of Vector Addition

Study the following example of the component method of vector addition.

Example problem: Add the following three vectors using the component method: A = 10.00 m, 60°; B = 6.00 m, 135°, C = 3.00 m, 270°. Determine the magnitude and direction angle of the sum.

Solution:

Organize the components into a table like the one below. Then add the components of the individual vectors to find the components of the vector sum.

Vector X-comp
(m) Y-comp
(m)

A 5.00 8.66

B -4.24 4.24

C 0.00 -3.00

Sum, D 0.76 9.90

Find the magnitude of the vector sum. The vector sum is also known as the resultant.
The direction angle of the resultant is the following. Note that the angle must be a Quadrant I angle, because both x- and y-components of the sum are positive.

Carry out the method shown above for problem 3-20 of the text.

Part B. Graphical Method of Vector Addition

In the graphical method, the individual vectors are placed tip-to-tail. Then the resultant is drawn from the tail of the first vector to the tip of the last vectors. You've already had some experience with this from E.3.2. In that assignment, you drew freehand sketches. In the present assignment, you'll do a careful, scaled construction on graph paper. Here are the specifics.

Download and print a sheet of graph paper. You'll construct your vector drawing on this page and include the page in the file that you submit.
In constructing your drawing, do the following:
1. Make the drawing large. Remember that greater measurements generally mean greater accuracy.
2. Draw the vectors to scale. Select a convenient scale factor, say 1.0 major division (5 minor divisions) on the graph paper represents 1.0 m of distance. In that case, vector A would be drawn 10.0 major divisions long.
3. Use a protractor to measure angles. Use a ruler to measure lengths and draw the vectors straight.
4. Put arrowheads on the ends of the vectors. Otherwise, they're just lines. Label the vectors A, B, C.
5. Draw the vector sum A + B + C from the tail of A to the tip of C. Label the vector D.
6. Measure the length of D with your ruler. Then use your scale factor to convert back to meters.
7. Measure the direction angle of D with your protractor. Show clearly on your scale drawing the angle that you measured.
When you've completed the above, compare the magnitudes and angles found with the component and graphical methods. That means you need to say quantitatively how close they are to each other. The way to do this is to calculate the percentage difference:

% Difference = 100 ∙ (Value 1 - Value 2) ÷ (Value 1 + Value 2)