Resultant of Three Concurrent Coplanar Forces

Cornelis Kok
10 Aug 201611:18
EducationalLearning
32 Likes 10 Comments

TLDRThis script demonstrates how to calculate the resultant force for a concurrent, coplanar system of three forces acting at a single point O. By resolving each force into its X and Y components using trigonometric functions, the script shows how to find the individual components and then sum them up to get the total X and Y components. The resultant force is determined using the Pythagorean theorem and the angle of this force is calculated using the tangent function. The final answer is a resultant force of 10.8 Newtons at an angle of negative 18.5 degrees from the positive X-axis, providing a clear and methodical approach to force analysis.

Takeaways
  • 📐 The script explains how to calculate the resultant force for a concurrent, coplanar system of forces.
  • 🔢 Three forces are considered acting at a single point O with specified magnitudes and directions.
  • 🧭 Each force is broken down into its X and Y components using trigonometric functions based on their angles from the X-axis.
  • 🌟 The X and Y components are calculated using cosine and sine of the respective angles (45, 150, and 300 degrees).
  • 📈 The resultant force is found by summing the individual X and Y components of the forces.
  • 🤹 The tabular method is introduced as a systematic way to compute the resultant force components.
  • 📊 The resultant force magnitude is determined using Pythagoras' theorem with the summed components.
  • 📐 The direction of the resultant force is calculated using the tangent function based on the X and Y components.
  • 🎯 The final answer includes both the magnitude (10.8 Newtons) and the direction (negative 18.5 degrees from the positive X-axis) of the resultant force.
  • 🔗 A video review is mentioned for those needing a refresher on finding force components.
  • 📝 The process is explained step by step, making it educational and easy to follow for the audience.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is calculating the resultant force for a concurrent, coplanar system of forces.

  • How many forces are acting at the single point O?

    -There are three forces acting at the single point O.

  • What are the magnitudes and directions of the three forces?

    -The magnitudes and directions of the three forces are: a 10 Newton force at 45 degrees from the horizontal, a 5 Newton force at 30 degrees from the negative x-axis, and a 15 Newton force at 60 degrees from the positive x-axis.

  • How can the forces be resolved into their X and Y components?

    -The forces can be resolved into their X and Y components by using the cosine and sine of the respective angles from the x-axis. For example, the X component of a force is the magnitude multiplied by the cosine of the angle, and the Y component is the magnitude multiplied by the sine of the angle.

  • What is the total X component of the forces?

    -The total X component of the forces is 10.24 Newtons, calculated by summing the individual X components (10 cos 45°, 5 cos 150°, and 15 cos 300°).

  • What is the total Y component of the forces?

    -The total Y component of the forces is -3.42 Newtons, calculated by summing the individual Y components (10 sin 45°, 5 sin 150°, and 15 sin 300°).

  • How is the resultant force determined?

    -The resultant force is determined by combining the X and Y components. It is the vector sum of the individual forces and can be found using the Pythagorean theorem: R = √(X component² + Y component²).

  • What is the magnitude of the resultant force?

    -The magnitude of the resultant force is approximately 10.8 Newtons.

  • How is the angle of the resultant force measured?

    -The angle of the resultant force is measured from the positive x-axis and is determined by using the tangent function: θ = arctan(opposite/adjacent). In this case, θ = arctan(-3.42/10.24), which is approximately -18.5 degrees.

  • What is the significance of the negative sign in the Y component?

    -The negative sign in the Y component indicates that the force is acting in the downward direction along the y-axis.

  • How can the tabular method help in finding the resultant force?

    -The tabular method helps in systematically organizing and calculating the individual X and Y components of each force, making it easier to find their sums and, ultimately, the resultant force.

Outlines
00:00
📐 Introduction to Concurrent Coplanar Force Analysis

This paragraph introduces the concept of calculating the resultant force in a concurrent coplanar system of forces. It describes a scenario with three forces acting at a single point O - a 10 Newton force at 45 degrees from the horizontal, a 5 Newton force at 30 degrees from the negative x-axis, and a 15 Newton force at 60 degrees from the positive x-axis. The paragraph explains the process of resolving these forces into their X and Y components using trigonometric functions, setting the stage for further analysis.

05:03
🔢 Calculation of Resultant Force Components

The second paragraph delves into the method of calculating the resultant force by summing the X and Y components of the individual forces. It provides a step-by-step explanation of how to find the total X components and Y components by adding the respective values obtained from the cosine and sine of the angles. The paragraph also introduces the tabular method for a clearer understanding and presents the calculations for the X and Y components, resulting in a total X component of 10.24 Newtons and a Y component of -3.42 Newtons. The negative sign for the Y component indicates the direction towards the negative Y-axis.

10:07
📊 Visualization and Final Result of Force Analysis

The final paragraph focuses on visualizing the resultant force and calculating its magnitude and direction. It describes how to represent the X and Y components as vectors and how to use these to determine the resultant force using Pythagoras' theorem. The magnitude of the resultant force is calculated to be approximately 10.8 Newtons. The paragraph also explains how to find the angle at which the resultant force acts using the tangent function, with the angle (theta) being negative 18.5 degrees from the positive x-axis. The summary of the resultant force is presented, encapsulating the magnitude and direction, providing a comprehensive conclusion to the force analysis.

Mindmap
Keywords
💡Resultant Force
Resultant Force refers to the single force that can replace a system of multiple concurrent forces acting at a point, without changing the overall effect on the object. In the video, the calculation of the resultant force demonstrates how to combine three individual forces into one, to understand the net effect on the object at point O.
💡Concurrent Forces
Concurrent forces are forces that act at a single point or along a common line on an object. In the context of the video, the three given forces are concurrent as they all act at point O, which allows for the application of vector addition principles to find the resultant force.
💡Coplanar System
A coplanar system refers to a set of vectors or forces that lie within the same plane. In the video, the coplanar nature of the forces is implicit as all the forces are described as acting within the same coordinate system, with angles measured from the same reference axis.
💡Force Components
Force components refer to the individual horizontal (X) and vertical (Y) parts of a force. In the video, the forces are resolved into their X and Y components using trigonometric functions, which is essential for calculating the resultant force.
💡Trigonometric Functions
Trigonometric functions, such as cosine and sine, are mathematical functions that relate the angles and sides of a right-angled triangle. In the context of the video, these functions are used to calculate the X and Y components of the forces based on their magnitude and the angles they make with the reference axis.
💡Vector Addition
Vector addition is the process of combining two or more vectors to find a resultant vector that has the same effect as the original vectors. In the video, vector addition is used to combine the individual forces into a single resultant force by adding their components along the X and Y axes.
💡Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the video, the theorem is used to calculate the magnitude of the resultant force from the X and Y components.
💡Angle of Force
The angle of force refers to the direction in which a force is acting relative to a reference direction, often measured from a horizontal axis. In the video, the angle of the resultant force is calculated using the tangent function, which provides the direction of the net force.
💡Newton
Newton is the SI unit of force. It is defined as the force required to accelerate a one-kilogram mass by one meter per second squared. In the video, all forces and the resultant force are measured in Newtons, which quantifies the magnitude of the forces acting on the system.
💡Diagram
A diagram is a visual representation or a sketch used to illustrate the components or relationships of a system. In the video, a diagram is used to visually represent the forces and their components, as well as to show the direction and magnitude of the resultant force.
Highlights

The example demonstrates calculating the resultant force for a concurrent, coplanar system of forces.

Three forces act at a single point O, with specified magnitudes and directions.

The 10 Newton force makes a 45-degree angle from the horizontal x-axis.

The 5 Newton force acts 30 degrees from the negative x-axis.

The 15 Newton force makes a 60-degree angle from the positive x-axis.

Forces are resolved into their X and Y components using trigonometric functions.

The X component for the 10 Newton force is calculated as 10 cosine 45 and Y as 10 sine 45.

The X component for the 5 Newton force is calculated as 5 cosine 30 and Y as 5 sine 30.

The X component for the 15 Newton force is calculated as 15 cosine 300 and Y as 15 sine 300.

The total X component is found by summing the individual X components.

The total Y component is similarly found by summing the individual Y components.

The resultant force is determined by combining the X and Y components using vector addition principles.

The magnitude of the resultant force is calculated using the Pythagorean theorem.

The direction of the resultant force is found using the tangent function.

The final resultant force is 10.8 Newtons at an angle of negative 18.5 degrees from the positive x-axis.

The method can be applied to various practical scenarios involving force analysis.

Transcripts

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