The X component of the resultant is the sum of the X components of the
vectors being added, and similarly for the Y component.
The angle that the resultant makes with the X axis is given by
and the magnitude is given by
Vector Addition
Polygon method - Vectors may be added graphically by repositioning
each one so that its tail coincides with the head of the previous one (see
figure. 2). The resultant (sum of the forces) is the vector drawn from
the tail of the first vector to the head of the last. The magnitude (length)
and angle of the resultant is measured with a ruler and a protractor, respectively.
Note: In order to measure the angle, a set of axes must first be defined.
Figure 2 Vector addition by the polygon method
Component method - Vectors may be added by selecting two perpendicular
directions called the X and Y axes, and projecting each vector on to these
axes. This process is called the resolution of a vector into components
in these directions. If the angle a that the vector makes from the
positive X axis, is used (see figure 3), these components are given by
Figure 3 Finding the two perpendicular components of a vector.
Thus, if four forces act on an object at rest, the following relationship has to be satisfied.
An equivalent statement is
so thatis equal in magnitude and opposite in direction to the resultant of the other three forces.
Procedure:
Set up the following situations so that in each case the magnitudes of the forces are unequal.
1. Attach three strings about 12 cm long to the small washer and connect the other end to spring balances, to the end connected directly to the center force indicating shaft. Connect string loops, about 8 cm long, to the other end of the spring balances and wrap these loops around the 1-2 kg weights (see figure 4). You will need to make sure that when there is no load on the spring balance the scale reads zero. If it does not, you will need to adjust it by sliding the metal tab at the top of the device.
a) Move the weights so that the angle between forces F_{1} and F_{2} is 90 (see figure 5). On a paper (as large as .3 by .3 m, if possible) draw lines parallel to its edges and intersecting near its center. These lines will act as the X and Y axes, described in the Physical Principles section. Position the paper so that the origin of the axes is right under the small washer, with the forces F_{1} and F_{2 }along the two lines. Tape the sheet of paper to the table. Use a pencil to mark two points at opposite ends of the string supplying the force F_{1}. By connecting these two points, draw a line below the string showing the direction of the force. Following the same procedure, draw the direction of the other two forces. Record the weight on each string in Newtons. For those spring balances calibrated in grams, convert the scale readings by multiplying by 9.80*10^{-3} N/g. Place arrows on your lines in the direction of the force exerted by the spring balances. Select your X axis to be along the line of force F_{1}. Add the vectors for F_{1} and F_{2} both graphically (polygon method) and with trigonometry (component method). Compare the magnitude of the resultant with that of the force, F_{3} for both solutions. Using a protractor, measure a_{3} and compare it with the similarly measured angle of your graphical addition and your trigonometrically computed angle. Do your measurements satisfy the requirements of Newton's second law?
b) Repeat as outlined in part (a) using the component method only, but with the angle between F_{1} and F_{2} at about 120. Do your measurements satisfy the requirements of Newton's second law?
2. Repeat step 1a, using only the component addition method with 4 spring balances (see figure 6). Draw the forces F_{1}, F_{2}, F_{3}, and F_{4} approximately as illustrated. Find and add the components of F_{1}, F_{2}, and F_{3}. Compute the magnitude and direction of the sum of these forces and compare your result with a_{4} and F_{4}. Do your measurements satisfy the requirements of Newton's second law?
3. If time permits, for extra credit, repeat as in step 2 using 5 forces extended approximately as illustrated in figure 7. Do your measurements satisfy the requirements of Newton's second law?
Recording data:
Force | Magnitude (N) | Angle () |
Force 1 | ||
Force 2 | ||
Force 3 | ||
Resultant of 1 & 2 |
Table 2 Component Method
Direction | Force 1 | Force 2 | Resultant |
X | |||
Y |
Magnitude of resultant = _______________________
Angle of resultant = ___________________________
Direction | Force 1 | Force 2 | Resultant |
X | |||
Y |
Magnitude of resultant = _______________________
Angle of resultant = ___________________________
Direction | Force 1 | Force 2 | Force 3 | Resultant |
X | ||||
Y |
Magnitude of resultant = _______________________
Angle of resultant = ___________________________
Direction | Force 1 | Force 2 | Force 3 | Force 4 | Resultant |
X | |||||
Y |
Magnitude of resultant = _______________________
Angle of resultant = ___________________________