Calculating the Resultant Force and its Angle: A Case Study

In this detailed article, we explore a common physics problem involving the calculation of the resultant force when two forces are applied at an angle to each other. This problem, concerning a 4N and 5N force with an angle of 60 degrees between them, is a fundamental concept in both physics and engineering. We will solve this problem step-by-step, using vector components and trigonometric methods, and provide a practical calculator approach for verification.

Problem Description

We are given two forces: F1 5 N at 0 degrees, and F2 4 N at 60 degrees. The task is to find the magnitude of the resultant force (R) and its angle with respect to the 5N force (F1).

Solution Using Vector Components

Step 1: Determine the Components of F2

The force F2 can be broken down into its components along the reference direction (defined by F1) and the direction perpendicular to it:

Component along the reference direction: 4cos60 2 N Component perpendicular to the reference direction: 4sin60 2√3 N ≈ 3.46 N

Step 2: Calculate the Total Components

Adding these components to the reference force F1 (5 N at 0 degrees), we get the total components:

Total component along the reference direction: 5 N 2 N 7 N Total component perpendicular to the reference direction: 3.46 N

Step 3: Calculate the Resultant Force

The resultant force can be found using the Pythagorean theorem:

R √(7^2 3.46^2)

R ≈ √(49 11.97) √60.97 ≈ 7.81 N

The angle φ with respect to F1 is:

φ tan^(-1)(3.46/7) ≈ 26.3 degrees

Verification Using Trigonometric Methods

Method 1: Using the Cosine Rule

The resultant force R can also be found using the cosine rule:

R2 F12 F22 - 2 × F1 × F2 × cos(180 - 60)

R2 52 42 - 2 × 5 × 4 × cos(120)

F2 × cos(120) -2 × 5 × 4 × (-0.5) 20

R2 25 16 20 61

R √61 ≈ 7.81 N

Method 2: Using Trigonometric Functions

Using the sine theorem, we can find the angle of the resultant with respect to the 5N force:

sin X (4sin120) / √61

sin X (4 × 0.866) / 7.81 ≈ 0.433

X sin^(-1)(0.433) ≈ 25.6 degrees

Using a Calculator for Verification

For verification, we can use a TI-84 calculator. First, change the MODE from REAL to re^θi and store πi/180 into variable w to convert degrees into radians.

Input 4e^θi0w5e^θi60w The calculator should return the resultant force approximately as: 7.810249676e^θi33.67049651i, which is around 7.81 N at an angle of 33.67 degrees.

Frequently Asked Questions

Q: How do I solve similar problems?

A: Use vector components and the cosine rule or trigonometric functions. Always ensure to convert degrees to radians on calculators that require it.

Q: What is the difference between the angle found and the resultant angle?

A: The angle found is the angle between the resultant force and the 5N force, while the resultant angle is the direction of the resultant force from the reference direction.

Q: Why is the cosine rule preferred in some cases?

A: The cosine rule is preferred when the angle between the forces is greater, as it simplifies the calculation compared to trigonometric methods for larger angles.