Calculating the Resultant Force: An Example with Forces Acting at Angles

Introduction

In physics, forces can be represented using vectors. Understanding how to calculate the resultant force when multiple forces act at different angles is a fundamental concept. This article provides a detailed explanation using a specific example where two forces, F1 and F2, act at 30° and 60° respectively.

Problem Statement

Given two forces, F1 10 N at θ1 30° and F2 20 N at θ2 60°, we will find the resultant force using vector decomposition.

Step-by-Step Solution

Step 1: Resolve Each Force into Components

First, we resolve each force into its components along the X and Y axes.

F1

F1x F1 cos(30°) 10 cos(30°) 10 × (sqrt{3}/2) ≈ 8.66 N F1y F1 sin(30°) 10 sin(30°) 10 × (1/2) 5 N

F2

F2x F2 cos(60°) 20 cos(60°) 20 × (1/2) 10 N F2y F2 sin(60°) 20 sin(60°) 20 × (sqrt{3}/2) ≈ 17.32 N

Step 2: Sum the Components

Next, we sum the X and Y components to find the resultant force's components.

Total X-component:

FRx F1x F2x 8.66 N 10 N 18.66 N

Total Y-component:

FRy F1y F2y 5 N 17.32 N 22.32 N

Step 3: Calculate the Magnitude of the Resultant Force

Using the Pythagorean theorem, we can find the magnitude of the resultant force.

R sqrt{FRx2 FRy2} sqrt{18.662 22.322}

Calculating:

R ≈ sqrt{348.6596 498.3424} ≈ sqrt{847.002} ≈ 29.14 N

Step 4: Calculate the Angle of the Resultant Force

Finally, we find the angle of the resultant force using the arctangent function:

.phi tan -1 (FRy / FRx) tan -1 (22.32 / 18.66) ≈ 50.19°

Conclusion

The resultant force is approximately 29.14 N at an angle of 50.19° from the X-axis.

Alternative Methods

Additionally, the resultant force can be calculated using the cosine law and the law of sines:

Cosine Law

R sqrt[102 202 - 2 × 10 × 20 cos(150°)] sqrt[100 400 - 200(-0.866)] ≈ 29.093 N

Using the law of sines, the angle α of the resultant force with respect to the 10 N force can be found:

29.093 / sin(150°) 20 / sin(α) → sin(α) (20 × sin(150°)) / 29.093 → α ≈ 36.87°

Using a TI-84 Calculator

With a TI-84 calculator:

Store πi/180 in w to convert degrees to radians. Input 10e30w 20e45w. This results in approximately 29.77197223e40.01276527i, which is 29.77 N at an angle of 40°.

Summary

In summary, the resultant force can be accurately calculated by decomposing the forces into their components, summing the components, and then applying the Pythagorean theorem and trigonometric functions.