How do you find the resultant of a coplanar concurrent force system by the method of resolution?

In one line

Add up all horizontal components and all vertical components separately; these two sums give the size and direction of the single resultant force.

Explained simply

Imagine five people pulling a ring tied with five ropes in different directions. To predict where the ring will move, list how much each person pulls right-left and up-down, add them up, and you get one net pull. Because all forces pass through the ring, there is no turning effect.

Technically, for coplanar concurrent forces the resultant passes through the point of concurrency and is fully defined by \( \Sigma F_x \) and \( \Sigma F_y \).

How it works

1. Draw the forces from the common point and choose x and y axes (right and up positive).
2. For each force, write \( F\cos\theta \) and \( F\sin\theta \) with correct signs (a table helps).
3. Add: \( \Sigma F_x \) and \( \Sigma F_y \).
4. Magnitude \( R = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2} \).
5. Direction \( \theta = \tan^{-1}|\Sigma F_y/\Sigma F_x| \), then fix the quadrant from the signs of the sums.
6. Draw R on a final sketch through the common point.

Key formula

\( R = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2} \), \( \theta = \tan^{-1}\left|\dfrac{\Sigma F_y}{\Sigma F_x}\right| \). Forces in N, R in N, \( \theta \) measured from the x-axis in the quadrant given by the signs.

Quick example

Forces: 20 N along +x, 30 N at 90° (along +y), 10 N at 180°. \( \Sigma F_x = 20 - 10 = 10 \) N, \( \Sigma F_y = 30 \) N. \( R = \sqrt{100 + 900} = 31.62 \) N, \( \theta = \tan^{-1}(3) = 71.57^\circ \) in the first quadrant (both sums positive).

Common mistakes

• Reporting the calculator angle without checking the quadrant; if \( \Sigma F_x \) is negative and \( \Sigma F_y \) positive, R is in the 2nd quadrant.
• Taking moments in a concurrent system; it is unnecessary since all forces pass through one point.
• Mixing angles measured from different axes in the same table.

How MU asks it

Very common 5 to 6-mark numerical: 'Find the magnitude and direction of the resultant of the forces shown'. Sometimes reversed: 'find the unknown force P and its angle so that the resultant is ... N along ...'. For full marks show the resolution table, sums with units, R, the angle with quadrant, and a final sketch of R.