Resultant Force Calculator

Understanding Resultant Force Calculator

The resultant force is the single force that has the same effect as all forces acting on an object combined. For perpendicular forces, it's the Pythagorean theorem.

How to use this calculator

1. Enter the first force as \(F_1\).
2. Enter the second force as \(F_2\).
3. Enter the angle \(\theta\) between the two forces.
4. Click calculate to get the resultant force \(R\).

The formula explained

The formula \(R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta}\) computes the magnitude of the single resultant force from two forces acting with an included angle. It uses the law of cosines to combine both force sizes and their direction difference.

• R = the resultant force
• \(F_1\) = the first force
• \(F_2\) = the second force
• \(\theta\) = the angle between the two forces

Step by step method

1. Square each force, so find \(F_1^2\) and \(F_2^2\).
2. Find the angle term \(2F_1F_2\cos\theta\), then add all three parts.
3. Take the square root of the total to get \(R\).

Worked example

Problem. Find the resultant force when \(F_1 = 10\,\text{N}\), \(F_2 = 15\,\text{N}\), and \(\theta = 60^\circ\).

1. Substitute into the formula, \(R = \sqrt{10^2 + 15^2 + 2(10)(15)\cos 60^\circ}\).
2. Compute each part, \(R = \sqrt{100 + 225 + 300(0.5)} = \sqrt{475}\).
3. Take the square root, \(R \approx 21.8\,\text{N}\).

Tips and common mistakes

• Make sure the angle is the angle between the two forces, not an angle from a triangle unless it is the included angle.
• If \(\theta = 0^\circ\), the forces add, and if \(\theta = 180^\circ\), they oppose each other, so the result can be much smaller.

Frequently asked questions