Question

How do you find the resultant of coplanar concurrent force system?

Answer 1

 The resultant force, of a given system of forces may be found out by the method of resolution as discussed below:

Let the forces be P₁, P₂, P₃, P₄, and P₅ acting at ‘o’. Let OX and OY be the two perpendicular directions. Let the forces make angle a₁, a₂, a₃, a₄, and a₅ with Ox respectively. Let R be their resultant and inclined at angle θ. with OX

Resolved part of ‘R’ along OX = Sum of the resolved parts of P₁, P₂, P₃, P₄, P₅ along OX.

i.e., Resolve all the forces horizontally and find the algebraic sum of all the horizontally components (i.e., ∑H)

Rcosθ = P₁cosα₁ + P₂cosα₂ + P₃cosα₃ + P₄cosα₄ + P₅cosα₅ = X (Let)

Resolve all the forces vertically and find the algebraic sum of all the vertical components (i.e., ∑V) 

Rsin? = P₁sinα₁ + P₂sinα₂ + P₃sinα₃ + P₄sinα₄ + P₅sinα₅ = Y (Let)

The resultant R of the given forces will be given by the equation:

R = √ (∑V)² + (∑H)²

We get R²(sin²θ + cos²θ) = P₁²(Sin²α₁+ cos²α₁) + ---

i.e., R₂ = P₁² + P₂² + P₃² + ---

And The resultant force will be inclined at an angle ‘θ’with the horizontal, such that

tanθ = ∑V/∑H

NOTE:

1. Some time there is confusion for finding the angle of resultant (θ), The value of the angle θ will be very depending upon the value of ∑V and ∑H, for this see the sign chart given below, first for ∑H and second for ∑V.

a. When ∑V is +ive, the resultant makes an angle between 0º and 180º. But when ∑V is –ive, the resultant makes an angle between 180° and 360°. 

b. When ∑H is +ive, the resultant makes an angle between 0º and 90° and 270° to 360°. But when ∑H is -ive, the resultant makes an angle between 90° and 270°.

2. Sum of interior angle of a regular Polygon = (2.n – 4).90°

Where, n = Number of side of the polygon

For Hexagon, n = 6; angle = (6 X 2–4) X 90 = 720°

And each angle = total angle/n = 720/6 = 120° 

3. It resultant is horizontal, then θ = 0º

i.e. ∑H = R, ∑V = 0

4. It Resultant is vertical, then θ = 90º; i.e., ∑H = 0, V = R