What is the resulant of concurrent forces?

Question

for marirme students

Answer 1

When we talk about resultant of forces we are talking of vector quantities, which are quantities that have both magnitude and direction.

To get the resultant of concurrent forces, perform the following:

1. Arrange the forces so that they are all pointing to the point of intersection or origin.

2. Multiply the magnitude of each force by the sine of its angular direction. The angular direction is measured from the right side of the x- axis in a counter clockwise direction. Remember that the sign of angles are positive in the first and second quadrant and negative in the third and last quadrant.

3. Take the algebraic sum of the products obtained from step 2. Express this as y. Mathematically this can be represented as;

y = a sinA + bsinB  + c sin C +.....

Where the small letters are the magnitude of the forces and the capital letters are the corresponding angular directions of the forces

3. Multiply the magnitude of each force with the cosine of its angular direction. Rememember that the cosine of angles are positive in the first and fourth quadrant, while they are negative in the second and third quadrant.

4. Add the products obtained from step 3 algebraically. Denote the sum as x. This can be mathematically represented as;

x = acosA + bcosB + c cosC +.....

5.  Obtain the resultant of the forces by;

R = sqrt (x^2 + y^2)

The resultant will still be ponting to the origin or point of intersection.

6. To locate the direction of the force, use x and y as the coordinates of the point defining the magnitude of the resultant. Hence;

Ar = arctan (x/y), wher Ar is the angular direction of the resultant.(use degrees for anglular measure unless you are more adept at using radian measure). So as not to complicate matters disregard the signs of x and y just for this one so that Ar is taken as positive.

7. Now, to get exactly where the resultant is located. Look at the signs of x and y:

a. (x,y) means that the resultant is in the first quadrant so the angular direction is Ar.

b. (-x, y) means that the resultant is in the second quadrant. So the angular direction is (180 -Ar) in degrees

c. (-x,-y) means that the resultant is in the third quadrant. So the angular direction is (180 + Ar)

d. (x,-y) means that the resultant is in the fourth quadrant, So the angular direction is (360 -Ar).

Answer 2

Graphically:

Take each force one by one in sequential order of angle and lay them end on end without disturbing the direction. (Arrowheads against starting points)

You will get a regular or an irregular polygon with the last side open

Join the last side with a straight line but reverse its direction

This is the resultant