Resultant force

When a system of several forces and several couple moments are acting on a rigid body, we can replace all of them by a unique force and unique couple momemt, at a certain point, that will have the same overall external effect on the body. The obtained force and couple moment are called equivalent system.

1. Moving a force on its line of action

Since the body is rigid, moving a force along with its line of action, from the point A to the point O, which both are on the vector's line of action, does not change the external effect. Therefore this force vector is called a sliding vector. There is an internal effect within the body due to the force, that depends indeed on where this force is applied.

2. Moving a force of its line of action

Moving a force out off its line of action, from the point A to the point B, which both are not on the vector's line of action, does not change the external effect. Therefore this delocalized force is accompanied with a brand new couple moment. This moment is a free vector and can be applied at any point on the body.

3. Resultant force and resultant moments of a system

With several forces and their couple moments acting on a body, he rule is to move each force and its associated couple moment to a common point, and find the resultant force-couple moment. Therefore The total force is the sum of all the forces, and the total moment is the sum of the generated couple moments M_cg and the moments non generated M_ng by the forces:

At the common point:

Each force F_icomes to the point O and remains F_i plus a couple momemnt:
M_i = r_i x F_i
F_R = ΣF_i
M_RO = ΣMg_i + ΣMng_i

4. Reducing a force and couple system to a single force

If the resultant force and the resultant couple are perpendicular, then we can find an equivalent system with a single force and no couple.



To obtain this system, we move the resultant force a distance d along the line perpendicular to the plane of the resultant force and resultant couple until the resultant force creates a moment equivalent to the resultant couple: d = M_R/F_R

3. Independence of reference point: Free vector

Unlike the moment of a force that is always defined with respect to a certain reference point, the moment of a couple is independent of the reference point, hence it is a free vector.
Proof:

If we have a couple of force vectors F₁ and F₂, that form tho couple moments M₁ and M₂ with position vectors from a certain point r₁ and r₂, respectively. The couple moment about this point is:
M = M₁ + M₂ = r₁ x F₁ + r₂ x F₂.
Since F₁ = - F₂ = F, we get then:
M = (r₁ - r₂)x F.
The difference r₁ - r₂ is not a position vector, it is independent of point of reference.

Recall: The couple moment is a free vector.

4. Example

The rule is to:
- Sum all the x-components and y-components of the force to find the equivalent single force F_R,
- Sum all the moments with respect to the reference point O to finf the resultant moment M_RO , and
- Find the location of F_R, that is the distance d from the point O:

d = M_RO /F_R



A pivoted uniform boom with a cable is in equilibrium. The pivot at the wall exerts no torque.
The forces exerted on the boom lie in the x-y plane. These force are obtained using the scalar following equations:
F_Rx = ΣF_ix
F_Ry = ΣF_iy
M_RO = ΣMg_i
then obtainable from the force equation shown. The lever arm for the cable tension T must be obtained from the triangle as shown since the cable is not perpendicular to the boom.
F_x = 10 N, F_y = 12 N, T = 4 N
θ = 30^o
m1 = 2 kg, m = 1 kg, m2 = 3 kg
L = 4 m, x1 = 1 m, x2 = 3 m, x_i = 2.5 m

1. The expression of the resultant force:

F_Rx = ΣF_ix = F_Rx = F_x - T cos θ = 10 - 4 cos 30^o = 6.54 N

F_Ry = F_y + T sin θ -m1 g -m g - m2 g
= 12 + 4 sin 30^o - (2 + 1 + 3) 9.81 = - 44.90 N

F = [ F_Rx² + F_Ry²]^1/2 = 45.40 N

tg φ = -44.90/6.54 = -6.85
φ = - 82^o

1. The expression of the resultant moment:

We choose the point pivot O as the reference point, then:
M_RO = ΣMg_i = m1 x1 + m L/2 + m2 x2 - T sin θ x_T
= 1 x 1 + 1 x 2 + 3 x 3 - 4 (1/2) x 2.5 = 7 N.m

d = 7/45.40 = 0.15 m = 15.0 cm