Resultant Force: Formula, Examples and How to Find It

For any number of forces acting at a point, resolve them all into horizontal and vertical components first, then combine:

R = √((ΣFx)² + (ΣFy)²)

And the direction:

tan α = ΣFy / ΣFx

Where:

• R is the magnitude of the resultant force, in newtons (N)

• ΣFx is the algebraic sum of all horizontal components, in newtons (N)

• ΣFy is the algebraic sum of all vertical components, in newtons (N)

• α (alpha) is the angle of the resultant measured from the x axis, in degrees

The symbol Σ is the Greek letter sigma and simply means add them all up, keeping the signs.

Everything else in this guide is a special case of these two lines. Learn them properly and the rest becomes shortcuts rather than new rules.

Resultant of Forces in the Same and Opposite Directions

If two forces act along the same straight line:

• Same direction: R = F1 + F2, acting in that same direction

• Opposite directions: R = F1 − F2, acting in the direction of the larger force

Two people pushing a trolley the same way with 200 N and 150 N give 350 N forward. If one pushes and the other pulls back, you get 50 N in the direction of the stronger one.

Keep this useful limit in mind for every problem: the resultant of two forces can never be more than their sum or less than their difference. If you calculate a resultant of 400 N from a 100 N and a 150 N force, you have made an arithmetic mistake somewhere.

Resultant of Two Perpendicular Forces

When two forces meet at exactly 90 degrees, the components are already sorted for you and the formula shrinks to plain Pythagoras:

R = √(F1² + F2²) tan α = F2 / F1

Where α is the angle between the resultant and force F1, in degrees.

Solved example

A crate is pushed with 60 N horizontally while a rope lifts it with 80 N vertically.

1. R = √(60² + 80²) = √(3600 + 6400) = √10000 = 100 N

2. tan α = 80 / 60 = 1.333, so α = 53.1 degrees above the horizontal

The 3, 4, 5 triangle turning up here is not a coincidence. Examiners love those numbers, so if you spot a 3 to 4 ratio, you can often write the answer without a calculator.

One more shortcut worth remembering: if two perpendicular forces are equal, the resultant is always at 45 degrees, with magnitude F√2.

Resultant of Two Forces at Any Angle

When two forces meet at an angle that is neither 0, 90 nor 180 degrees, you can either resolve both into components or use the parallelogram law directly.

The parallelogram law of forces says that if two forces at a point are represented by the two adjacent sides of a parallelogram, their resultant is the diagonal drawn from the same point.

R = √(P² + Q² + 2PQ cos θ)

And the angle of the resultant from force P:

tan β = (Q sin θ) / (P + Q cos θ)

Where:

• P and Q are the two force magnitudes, in newtons (N)

• θ is the angle between the two forces, in degrees

• β (beta) is the angle between the resultant and force P, in degrees

Solved example

Two tugboats pull a ship with 5000 N and 4000 N, with 50 degrees between the ropes.

1. R = √(5000² + 4000² + 2 × 5000 × 4000 × cos 50°)

2. R = √(25000000 + 16000000 + 40000000 × 0.643)

3. R = √(25000000 + 16000000 + 25720000) = √66720000 = 8168 N

4. tan β = (4000 × sin 50°) / (5000 + 4000 × cos 50°) = 3064 / 7572 = 0.405

5. β = 22.0 degrees from the 5000 N rope

Check it against the limits. The sum is 9000 N and the difference is 1000 N, and 8168 N sits comfortably between them, so the answer is believable.

Resultant of Three or More Forces

Nothing new is needed here. The component method handles any number of forces without changing.

Resolve every force, add all the x components, add all the y components, then combine with the same formula. Ten forces take longer than two, but they do not take more thinking.

Solved example

Four forces act at a point: 50 N at 0 degrees, 80 N at 60 degrees, 40 N at 150 degrees and 60 N at 240 degrees.

1. Force 1: Fx = 50.0, Fy = 0

2. Force 2: Fx = 80 cos 60° = 40.0, Fy = 80 sin 60° = 69.3

3. Force 3: Fx = 40 cos 150° = −34.6, Fy = 40 sin 150° = 20.0

4. Force 4: Fx = 60 cos 240° = −30.0, Fy = 60 sin 240° = −52.0

5. ΣFx = 50.0 + 40.0 − 34.6 − 30.0 = 25.4 N

6. ΣFy = 0 + 69.3 + 20.0 − 52.0 = 37.3 N

7. R = √(25.4² + 37.3²) = √(645 + 1391) = √2036 = 45.1 N

8. tan α = 37.3 / 25.4 = 1.469, so α = 55.7 degrees from the x axis

Set your work out in a clear column of components. Most errors in multi-force problems come from losing track of a sign halfway down the page, not from the trigonometry.

The Graphical Way to See the Resultant

You can also find the resultant by drawing, and it is worth seeing once because it makes the idea physical.

Pick a scale, for example 1 cm = 20 N. Draw each force as an arrow of the right length and direction, joining them head to tail. Then draw one arrow from the very first tail to the very last head. That closing arrow is the resultant, and you measure its length and angle directly.

With two forces the shape is a triangle, which gives the triangle law of forces. With more forces it becomes a many-sided figure, which gives the polygon law of forces. Both say the same thing in different sized packages.

The most valuable insight comes from what happens when the drawing closes on itself. If the polygon closes with no gap, the resultant is zero. Engineers use that as a visual check on whether a joint is balanced before running a single calculation.

Use graphical work to build intuition and to sanity check. Use the component method when you need a number you would put your name to, because ruler accuracy is not design accuracy.

When the Resultant Force Is Zero

If the resultant of all forces on a body is zero, the body is in equilibrium. It will not start moving and it will not accelerate.

That condition gives the two equations behind almost all of statics:

ΣFx = 0 ΣFy = 0

Every support reaction, cable tension, truss member force and bolt load you will ever calculate comes out of those two lines, plus a third for moments in rigid body problems.

It is worth being precise about one thing. Zero resultant does not mean the body is standing still. It means the body is not accelerating. A car cruising at a steady 60 km/h has zero resultant force on it, because the engine thrust exactly cancels drag and friction.

Resultant Force and Acceleration

Statics is the case where the resultant is zero. The moment it is not zero, the body accelerates, and Newton's Second Law takes over:

R = ma

Where:

• R is the resultant force in newtons (N)

• m is the mass in kilograms (kg)

• a is the acceleration in metres per second squared (m/s²)

The acceleration always points in the same direction as the resultant, which is a useful sanity check. If your calculated resultant points up the slope but the box is clearly sliding down, something in your signs is wrong.

Solved example

A 40 kg trolley has a 200 N push forward and a 50 N friction force backward.

1. R = 200 − 50 = 150 N forward

2. a = R / m = 150 / 40 = 3.75 m/s² forward

Resultant Force and Equilibrant Force