Determine the Resultant of Two Forces Making Arbitrary Angles – Solved Problem

In this mechanics, physics, and statics tutorial, we explain how to find a resultant of two forces acting at a point. The two forces make arbitrary angles. We solve this problem by using three approaches. The first approach is based on applying the parallelogram method. The second approach is based on the triangle rule. The third approach is a numerical approach that is based on the application of the law of cosine and the law of sin. Every engineer and scientist need to known how to solve the problem explained in this tutorial. The YouTube video accompanying this tutorial is given below.

The problem is described below.

PROBLEM OF FINDING THE RESULTANT OF TWO FORCES

Two forces and (see the figure below). Determine their resultant (find the intensity of the resultant and angle).

We will present three approaches to solve this problem.

Geometrical Solution Using the Parallelogram Method

We can use the parallelogram method. The parallelogram method is explained in the figure below.

To apply this method, you need a ruler measuring the distance and a protractor measuring the angle, or you can perform this task on a computer by using vector drawing software. This is a purely experimental graphical method. In step 1, we construct a line parallel to the action line of the force . In step 2, we construct a line parallel to the action line of the force , and find the intersection point of the two parallel lines. The intersection point is denoted by . In step 3, we construct the resultant starting from the point and ending at the point . In step 4, we measure the length of the resultant and the angle .

The length of the resultant force is and the angle is .

Geometrical Solution Using the Triangle Rule

The triangle rule for determining the resultant is similar to the parallelogram rule explained above. 1

In step 1, without changing the orientation of the force , we translate the force such that the beginning of the force is placed at the end of the force . That is, the tail of the force is placed at the tip of the force . This is called the “tip-to-tail” connection. In step 2, we connect the start (tail) of force to the end (tip) of force . In this way, we construct the resultant force . In step 3, we measure the length of the resultant and the angle .

The length of the resultant force is and the angle is .

Trigonometric Solution

To be able to solve this problem, we need to use the law of cosines and the law of sines. Consider a triangle shown below.

The law of cosines states the following:

(1)  

The law of sines states the following:

(2)  

After revising the laws of cosines and sines, let us go back to our problem. Consider the resultant force triangle shown in the figure below.

We want to compute the intensity of the resultant force and the angle . For that, we are going to use the law of cosines. By applying the law of cosines, we obtain

(3)  

where , , and are the intensities of , , and forces. From the last equation, we obtain

(4)  

By inserting the values in a calculator or in a Python code (the code is given at the end), we obtain

(5)  

Here, it is important to keep in mind that while computing the intensity of the resultant force, we need to convert angles from degrees to radians since the cos function argument is expressed in radians in MATLAB and Python.

On the other hand, we compute the angle by computing the angle . We compute the angle by using the law of sines. Namely, from Fig. 5, we have

(6)  

From the last equation, we have

(7)  

Finally, we obtain

(8)  

By substituting the values for and , we obtain

(9)  

Finally, the angle is

(10)  

The Python code for computing the resultant force is given below.

import numpy as np

F1=30
F2=20

# angles in degrees

a1=20
a2=15


# convert to radians

a1rad= a1*(2*np.pi)/360
a2rad= a2*(2*np.pi)/360

R=np.sqrt(F1**2+F2**2+2*F1*F2*np.cos(a1rad))

# test, using the original angle equal to 160
Rtest= np.sqrt(F1**2+F2**2-2*F1*F2*np.cos((180-a1)*(2*np.pi)/360))


# compute the angle 

angleAlphaRad=np.arcsin((F1/R)*np.sin((180-a1)*(2*np.pi)/360))

angleAlpha=angleAlphaRad*360/(2*np.pi)

betaAngle=angleAlpha+a2