Vector Addition of Forces, by Marcus Wade

Purpose:

The purpose of Vector Addition of forces was to study vector addition both by graphing, and by using vector components with trigonometry.

Equipment:

Tools that were used in this lab included a protractor, string, pulleys, varying mass (g), and a circular force table.

Procedure:

Circular Force Table

In this lab, three vectors were given, each with their own magnitude (g), and direction (angle in degrees). using a scale of 1cm= 20 (g) of mass, a vector diagram was constructed in which three separate force vectors were added.

fig 1

First, Vector A=200g at 0 degrees is graphed. Shown to the right. At an angle of 0 degrees, it simply follows the x axis for 10 cm (fig 1).

fig 2

Second,  Vector B=100g at 41 degrees is added to Vector A. Since its magnitude is 100g, the length of the vector is 5cm (fig 2).

fig3

Third, Vector C=150g at 132 degrees is added to the other Vectors, with a length of 7.5cm (fig3).

fig 4

Last, the resultant Vector R is connected from the tail of the Vector A to the head of Vector C. from graphing the resultant vector, its length appeared to be 12.5cm or 250g, at roughly an angle of 45 degrees (fig 4).













In order to obtain a more accurate depiction of Vector R, the x and y components were calculated using trigonometry for the three vectors that were added. The cos and sin of the vectors are used in order to get the magnitude of the x and y component vectors of A, B and C.

fig 5

Components and angles are shown in fig 5.

Where A= angle of vector

Sin(A)= ((y component)/(magnitude of vector))

(magnitude of vector)(sin(A))=(y component)

and also

Cos(A)= ((x component)/(magnitude of vector))

(magnitude of vector)(Cos(A))=(x component)

Plugging in all of the vector data to the equation, the x component of each vector are added to find R_x, and the y components are also added to find R_y.

R_x=200+100cos(41)+150cos(132)

R_x=175.1

R_y=0+100sin(41)+150sin(132)

R_y=177.1

Using our new values of R_x=175.1 and R_y=177.1, the exact magnitude and angle of Vector R can now be calculated.

fig 6

(Magnitude of Vector R)=(175.1²+177.1²)^1/2

(Magnitude of Vector R)=249g

(Angle of Vector R)=tan^-1(177.1/175.1)

(Angle of Vector R)=45.3 degrees

Measurements of resultant vector are shown in fig 6.

fig 7

The negative of Vector R then needed to be taken in order to create equilibrium between the forces. To do this, the vector components are simply made negative, and then take the new angle (225 degrees). Lets call this Vector -R (fig 7).

Setup:

Once the components of all the vectors had been obtained, the vectors were to be physically plotted on the circular force table. On the first holder, start with a force of 200 g at 0 degrees, then add a force of 100 g at an angle of 41 degrees on the second holder. Continuing,the third vector was added with mass 150 g at 132 degrees on the third holder.

What happens when you place a mass on the fourth holder equal to the magnitude of the resultant vector? When Vector -R (which is equal in magnitude to the resultant vector, but opposite in direction) is plotted physically and placed on the fourth holder, it creates equilibrium between the forces on the circular force table.

 Pictured in fig 8 is the equilibrium resulting from addition of all four force vectors.

The vector addition was then checked on a website (http://phet.colorado.edu/en/simulation/vector-addition). We input all of our vector information to find the resultant vector, and it seemed to match very well with our data (fig 9).

fig 9

Conclusion:

In this lab, one was able to learn about the addition of vectors both graphically and through using components. Graphing the vectors seemed to be a reasonably accurate way to estimate the magnitude and direction, but using the vector components was able to give one a clear and precise answer to our resultant vector. Possible sources of error in this lab would include having to estimate a weight in grams to balance, or even not setting the angles precisely on the circular force table.