Vector Addition of Forces

Vector Addition of Forces

Purpose:  To study vector addition by:

1) Graphical means. 

2) Using their components by using trigonometric Calculations. 

(A circular force table is used to check our results.)

Equipment:  

• 1 Protractor
• Some String
• Various Mass Plates
• 4 Pulleys
• 1 Circular Force Table

Procedure:  (From Lab Sheet)

1.  Your instructor will give each group three masses in grams (which will
represent the magnitude of three forces) and three angles. Choose a scale of
 1 cm = 20 grams, make a vector diagram showing these forces, and
graphically find their resultant. Determine the magnitude (length) and
direction (angle) of the resultant force using a ruler and protractor. 

2. Make a second vector diagram and show the same three forces again.  Find
the resultant vector again, this time by components. Show the components of
each vector as well as the resultant vector on your diagram. Draw the force
(vector) you would need to exactly cancel out this resultant.

3. Mount three pulleys on the edge of your force table at the angles given
above. Attach strings to the center ring so that they each run over the pulley
and attach to a mass holder as shown in the figure below. Hang the
appropriate masses (numerically equal to the forces in grams) on each string.
Is the ring in equilibrium?  Set up a fourth pulley and mass holder at 180
degrees opposite from the angle you calculated for the resultant of the first
three vectors. Record all mass and angles. If you now place a mass on this
fourth holder equal to the magnitude of the resultant, what happens?  Ask
your instructor to check your results before going on.

Setup:

Our group started with magnitudes 200cm, 100cm, and 150 cm with degrees of 0, 41, and 132 respectively. This is the graph of the vectors given: 

1 cm = 20 g

     Vector D is the resultant force. The angle of vector D and the x-axis is 45 degrees.

The graph shows the x and y components of each vector as well.

Vector A) Ax = 200g                                       x = 200+100cos(41)+150cos(132) = 175.1g
                Ay = 0g                                            y = 0+100sin(41)+sin(132) = 177.1g
Vector B) Bx = 100cos(41) = 75.5g
                By = 100sin(41) = 65.6g                 R = 250g at 45 degree
Vector C) Cx = 150cos(132) = -100.4g          Rx = 250cos(45) = 176.8g
                Cy = 150sin(132) = 111.5g             Ry = 250sin(45) = 176.8g

This graph only shows the x and y components of each vector, and the resultant vector is shown going from the tail of the first to the head of the last:

1 cm = 20 g

Using

 A= angle of vector

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

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

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

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

(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

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 transposed to negative (opposite vectors) and then take the new angle (225 degrees) is found, denoted Vector -R.

     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 200g at 0 degrees, then add a force of 100g at an angle of 41 degrees on the second holder. Continuing,the third vector was added with mass 150g at 132 degrees on the third holder.

Question:  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.

Circular Force Table w/ all masses; Circular Force Table in Equilibrium

Top View of Circular Force Table, showing the Equilibrium created by each Vector

This is the picture from the simulation website. using our vectors, we got the same resultant vector:

Vector Check

Simulated at:

 http://phet.colorado.edu/en/simulation/vector-addition

Conclusion:
     In this lab, we were able to learn about the addition of vectors both graphically and with 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.