PHYSICS LAB 1

A

R

Triangle Method

An equivalent method of finding R is to place the vectors to be added "head to tail" (head of A to tail of B, Fig. l b). Vector arrows may be moved as long as they remain pointed in the same direction. The length and direction of the resultant is measured from the graph.

Figure 1b

Polygon Method

If more than two vectors are added, the head-to-tail method forms a polygon (Fig. 2). For four vectors, the resultant R = A + B + C + D is the vector arrow from the tail of the A arrow to the head of the vector D. The length (magnitude) and the angle of orientation of R can be measured from the diagram.

Figure 2

b. Analytical Methods

Triangle Method

B

A

C

α

γ

β

The magnitude of R in Fig. 3 can also be computed by using trigonometry.

The Law of Sines and the Law of Cosines are especially useful for this:

Law of Sines: A/Sin α = B / Sin β = C / Sin γ. (3-1)

Law of Cosines: C2 = A2 + B2 – 2AB Cos γ (3-2)

Figure 3

Method Of Components

A vector A can be written as a sum of two vectors Ax and Ay along the x and y axis respectively, as shown. We call them the components of vector A and are given by:

Ax = A cosθ i (3-3)

Ay = A sinθ j (3-4)

where θ is the angle between the vector A and the x axis.

In order to find the resultant vector R of a system of vectors A, B, C, etc…

we follow these steps:

a) Find the x and y components for each vector using the above equations.

i.e. find Ax, Bx, Cx ... and Ay, By, Cy ....

Remember they can be positive or negative depending on their direction.

b) Add up these components to get:

Rx = Ax + Bx + Cx + … (3-5)

Ry = Ay + By + Cy + … (3-6)

c) Now, the magnitude of R is : [Rx2 + Ry2] ½

The direction of R is : θ = tan -1 [Ry / Rx] where θ is the angle between R and x axis.

If θ > 0 then R is either in the 1st or 3rd quadrant

If θ < 0 then R is either in the 2nd or 4th quadrant.

PROCEDURE

It would be good if you can print pages 4 to 7, and draw the vectors on them. If that is not possible, draw the vectors on white sheets of paper using the scale mentioned here.

1) VECTOR RESOLUTION: Consider a force vector = 300 N at the 40º angle. Resolve this vector into its x- and y-components by the following methods:

A) Graphical : Make the X- and Y-axes. Use a scale of 30 N = 1.0 cm, and draw an arrow of appropriate length at 40º. Drop perpendiculars from the tip of the vector to the X- and Y-axes. Measure the lengths of these lines and hence find the magnitudes of Fx and Fy. (do not calculate using trigonometry) Record the results.

B) Analytical : Compute the magnitudes of Fx and Fy by using the Component Method (equations 3-3 and 3-4). Record the results.

2) VECTOR ADDITION: Find the sum of two ‘forces’: 100 N at 30º and 200 N at 120º, by:

A) Graphical Method: Make the X- and Y-axes. Use a scale of 25 N = 1.0 cm. Draw arrows of appropriate lengths at 30º and 120º, both starting from the origin. Add them using the Parallelogram method. Measure the length and angle of the resultant. Convert length to magnitude of the resultant vector using the scale used to draw the vectors. Record the results in the Data Table.

B) Analytical Method: Compute the sum of the two vectors by using the Component Method (equations 3-3 and 3-4) as well as the Triangle Method (equations 3-1 and 3-2). Record the results in the Data Table.

3) VECTOR ADDITION: In addition to the two forces of procedure 2, add a third force = 150 N at 230º. Find the vector sum by A) Graphical and B) Component methods, and record in the Data Table. Use a scale of 25 N = 1.0 cm for the graphical method.

CALCULATIONS

Show your calculations for the analytic method and the method of components.

RESULTS

Write the results in the Table of Results on page 7.

ADDITIONAL INFORMATION:

http://www.physicsclassroom.com/class/vectors/u3l1b.cfm

http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html

https://www.youtube.com/watch?v=ZYbX8cL5LNE

OL-03 Addition of Vectors REPORT FORM