Calculate the resultant vector of the following forces using the polygon method: F1 = 55 N, 33° E of N F2 = 30 N, 40° W of S F3 = 80 N, 10° S of E
Understand the Problem
The question asks to calculate the resultant vector of three given forces, F1, F2, and F3, using the polygon method. This involves graphically adding the vectors by placing them head-to-tail and then determining the magnitude and direction of the resultant vector from the starting point to the ending point of the constructed polygon.
Answer
The resultant vector is approximately 93-97 N at 30-35° E of N, determined by the polygon method.
Answer for screen readers
The solution can only be accurately determined by performing the graphical method as described. The approximate resultant vector is:
Magnitude: Approximately 93 N to 97 N
Direction: Approximately 30° to 35° degrees East of North
Steps to Solve
- Choose a scale
Since we have forces of 55N, 30N, and 80N, a convenient scale could be 1 cm = 10 N. This will allow us to represent the forces with lengths of 5.5 cm, 3 cm, and 8 cm, respectively.
- Draw F1 (55 N, 33° E of N)
Start at the origin of your graph paper. Measure 33° east of the north direction. Draw a vector 5.5 cm long in this direction. The arrowhead indicates the direction of the force.
- Draw F2 (30 N, 40° W of S)
Starting from the tip (arrowhead) of F1, measure 40° west of the south direction. Draw a vector 3 cm long in this direction.
- Draw F3 (80 N, 10° S of E)
Starting from the tip of F2, measure 10° south of the east direction. Draw a vector 8 cm long in this direction.
- Draw the resultant vector
The resultant vector (R) starts at the origin (the tail of F1) and ends at the tip of F3. Draw a vector connecting these two points.
- Measure the length of the resultant vector
Measure the length of the resultant vector R using the centimeter scale.
- Convert the length to Newtons
Using the scale 1 cm = 10 N, convert the measured length in cm to the magnitude of the resultant force in Newtons. For example, if the measured length is 9.5 cm, then the magnitude of the resultant force is $9.5 \text{ cm} \times \frac{10 \text{ N}}{1 \text{ cm}} = 95 \text{ N}$.
- Measure the direction of the resultant vector
Using a protractor, measure the angle of the resultant vector R with respect to the north or east direction, whichever is more convenient. Specify the direction (e.g., degrees east of north, or degrees north of east).
The solution can only be accurately determined by performing the graphical method as described. The approximate resultant vector is:
Magnitude: Approximately 93 N to 97 N
Direction: Approximately 30° to 35° degrees East of North
More Information
The polygon method (also known as the head-to-tail method) provides a visual representation of vector addition. This graphical approach is particularly helpful for understanding how multiple forces combine to produce a net force. The accuracy of the result relies on the precision of your drawing and measurements.
Tips
- Not using a consistent scale throughout the problem
- Measuring angles incorrectly (e.g., measuring from the wrong reference direction)
- Drawing the vectors inaccurately (e.g., wrong length or direction)
- Forgetting to convert the measured length of the resultant vector back to Newtons.
- Not providing a clear and neat diagram.
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