Summing Inclined Forces: Laws of Cosines and Sines in Action

When dealing with forces that are inclined to each other, the resultant force and its direction need to be determined using specific mathematical principles. This article will explore how to find the resultant force and its direction using the laws of cosines and sines with a practical example.

Understanding Inclined Forces

In physics, when we have forces that are not in the same direction, their resultant can be determined using vector analysis. A common method involves using the laws of cosines and sines to find the magnitude (resultant force) and the direction.

General Method: Laws of Cosines and Sines

To find the resultant force, we use the law of cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles.

Step 1: Calculating the Resultant Force

Given the magnitudes of two forces and the angle between them, we can calculate the resultant force using the law of cosines. The steps are as follows:

Given:

Force $F_1 80 , text{N}$

Force $F_2 150 , text{N}$

Angle between forces $theta 60^circ$

Using the law of cosines:

$R^2 F_1^2 F_2^2 - 2 F_1 F_2 costheta$

Substituting the values:

$R^2 80^2 150^2 - 2 cdot 80 cdot 150 cdot cos60^circ$

$cos60^circ 0.5$

$R^2 6400 22500 - 2 cdot 80 cdot 150 cdot 0.5$

$R^2 6400 22500 - 12000$

$R^2 40900$

$R sqrt{40900} approx 202.24 , text{N}$

Step 2: Determining the Direction

To find the angle $phi$ of the resultant force with respect to $F_1$, we use the law of sines:

$frac{R}{sintheta} frac{F_2}{sinphi}$

Rearranging gives:

$sinphi frac{F_2 cdot sintheta}{R}$

Substituting the values:

$sinphi frac{150 cdot sin60^circ}{202.24}$

$sin60^circ frac{sqrt{3}}{2} approx 0.866$

$sinphi frac{150 cdot 0.866}{202.24} approx frac{129.9}{202.24} approx 0.642$

Now calculating $phi$:

$phi arcsin(0.642) approx 40.0^circ$

Summary

The resultant force $R approx 202.24 , text{N}$ is directed at an angle of approximately $40.0^circ$ from the direction of the 80 N force.

Note: This method relies on knowing the angle between the forces. If the angle is not known, the resultant force and its direction cannot be determined accurately.

Alternative Cases

If the two forces are in the same direction, their resultant is simply the sum of the magnitudes:

$R 80 , text{N} 150 , text{N} 230 , text{N}$

If the two forces are in opposite directions, the resultant is the difference between the magnitudes:

$R 150 , text{N} - 80 , text{N} 70 , text{N}$

The resultant force can vary between 70 N and 230 N depending on the angle between the two forces, which ranges from 0 to 180 degrees.

Practical Application

Consider a real-world example where you have two forces of 80 N and 150 N inclined at 60 degrees. This example demonstrates the practical application of the laws of cosines and sines to find the resultant force and its direction.

If you are working with forces that are not in the same or opposite directions, you need to know the angle between them to find the resultant force accurately.