Name: resolving forces in a plane
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 20 s
Channel: JoVE

力が x 軸に対して角度θで作用する 2 次元空間の構造要素を考えてみましょう。
力の作用線が原点を通過することを考慮すると、その成分はデカルト形式で表すことができます。力ベクトルの方向は、常にその成分の比率の逆数で与えられます。
これで、力ベクトルの作用線が原点を通過しない場合でも、そのベクトル成分はデカルト形式で表現できます。これらのベクトルコンポーネントの符号規則は、その方向に応じて選択できます。
ここでは、力は円周率からシータを引いた角度に向けられ、正のy軸から反時計回りに測定されます。
ここで、力の作用線が任意の角度アルファからベータを引いたものを、選択した座標系の正のx軸にする構造を考えてみます。力の成分は、同様の解析を使用して解決できます。

<ol>
	<li>メリアム、J.L.;Kraige、LGおよびBolton、JN(2020)。エンジニアリングメカニクス \u2012 静力学。ニュージャージー州ホーボーケン:ジョン・ワイリー。33ページ</li>
</ol>

二次元での力のシステム

機械工学における二次元でのシステムは、平面内での運動と力の解析を含みます。二次元の力ベクトルは、次のように成分に分解することができます：

<img alt="Equation 1" src="/files/ftp_upload/14229/14229_Equation_1v2.svg" style="text-align: center; height: 15px;" />

F_x と F_y は、F のベクトル成分であり、それぞれ x 方向と y 方向のベクトル成分です。これらのベクトル成分は、適切な単位ベクトルによるスカラー（F_x および F_y）の積で表すことができます。

直交座標系における力ベクトルの成分を求めるためには、まず力が正の x 軸となす角 θ を求める必要があります。力の作用線が原点を通過すると仮定すると、基本的な三角法を使用してその成分を直交座標系で表すことができます。

<img alt="Equation 2" src="/files/ftp_upload/14229/14229_Equation_2v3.svg" style="text-align: center; height: 34px;" />

ただし、力ベクトルの作用線が原点を通過しない場合でも、同じ手法を使用してベクトル成分を直交座標系で表すことができます。力ベクトルの方向は、その成分の比の逆接で与えられます。

<img alt="Equation 3" src="/files/ftp_upload/14229/14229_Equation_3v3.svg" style="text-align: center; height: 34px;" />

しかしながら、力ベクトルの作用線が原点を通過しない場合、そのベクトル成分は依然として同じ手法を使用して直交座標系で表すことができます。力ベクトルの方向に基づいて各成分の符号を選択することができます。力ベクトルをその成分に分解することにより、対象の構造体への力のネット効果を求めることができます。

二次元力システムの理解は、安全で構造的に安定した構造体の解析と設計をするためにエンジニアにとって重要です。この知識は、構造体がさまざまな力にどのように反応するかを基礎的に理解することを提供し、必要に応じて設計上の弱点を特定するのに役立ちます。

機械工学の二次元システムには、平面内の運動と力の分析が含まれます。2次元の力ベクトルを解くことができます...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Force Vectors

力のベクトル

KEY TERMS AND CONCEPTS

主要用語と概念

Two-Dimensional Force System

resolving forces in a plane

Resolving forces in a plane is a key part of two-dimensional mechanical analysis. A force vector can be split into x- and y-components, which show how the force acts in each direction. These components are written with the help of unit vectors, one for the x direction and one for the y direction.

To find the components in a Cartesian coordinate system, first identify the angle theta between the force and the positive x-axis. If the line of action passes through the origin, basic trigonometry gives the Cartesian form of the force. The magnitude of the force is F, and the direction can be found from the inverse tangent of the ratio of its components.

The same component method also works when the line of action does not pass through the origin. In that case, the sign of each component is chosen from the direction of the force vector. By breaking a force into parts, engineers can judge its net effect on a structure and look for possible weak points. This skill helps them analyze and design structures that are safe and structurally sound.

Resolving Forces in a Plane

scalar and vectors

Scalar vs Vector Quantities in Mechanics

Scalar and vector quantities are two main ways to describe physical quantities in mechanics. Common examples include force, speed, velocity, and work. Some of these are scalars, and others are vectors. A scalar is a quantity that has magnitude only. It does not need direction. Mass, area, and length are examples of scalar quantities. Scalars with the same units can be added or subtracted like ordinary numbers. A class that ends 10 min earlier than 50 min lasts 50 min minus 10 min, which is 40...

vector operations

Adding and Multiplying Vectors

Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses. A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and...

introduction to force

Force, Motion, and Mechanical Effects

Force changes the motion, shape, or orientation of an object. A flowing stream of water can push on a turbine vane and make it move in the direction of the flow. In mechanics, force is treated as a vector quantity, which means it has both magnitude and direction. It is measured in newtons, or N, in the SI unit system. Forces can act through direct contact or without contact. A push or pull is a contact force. Magnetic and gravitational forces are examples of non-contact interactions. For an...

force classification

Types of Forces in Physics

Forces in physics can be grouped by how they act and where they come from. These groups help describe motion, behavior, and equilibrium in the physical world. They are also useful in physics and engineering. One common way to classify forces is by whether they need contact. Contact forces act only when two objects touch. Frictional force, tension, and normal force are all contact forces that come from surface contact. Forces can also act without physical touch. These are called non-contact...

vector addition of forces

Resultant Force from Tugboats

Resultant force can be found with vector addition when two or more forces act on the same object. This idea helps explain how multiple pulls combine to create one net force. It is a key tool for understanding motion in physics. A clear example is a ship pulled by two small tugboats. The forces, F1 and F2, act at the same time but in different directions. The net force on the ship is the vector sum of those two forces. The parallelogram law gives a geometric way to add vectors. In this method,...

two-dimensional force system

Resolving forces in a plane is a key part of two-dimensional mechanical analysis. A force vector can be split into x- and y-components, which show how the force acts in each direction. These components are written with the help of unit vectors, one for the x direction and one for the y direction. To find the components in a Cartesian coordinate system, first identify the angle theta between the force and the positive x-axis. If the line of action passes through the origin, basic trigonometry...

two-dimensional force system: problem solving

two-dimensional-force-system-problem-solving

Two-Dimensional Forces: Free-Body Analysis

Two-dimensional force analysis starts with a free-body diagram. A free-body diagram shows the object and every external force acting on it. It includes each force’s size, direction, and point of application. After the diagram is drawn, each force is broken into x- and y-components. This uses vector analysis and trigonometry, especially sine and cosine. The forces can then be written in Cartesian vector form, which makes them easier to work with. Next, the components in each direction are...

scalar notation

Vector Components for Force Calculations

Scalar notation helps simplify vector calculations by breaking a vector into separate x and y parts. When vectors are added or subtracted, each component can be handled on its own using scalar notation. This makes force problems easier to solve.

cartesian vector notation

Vector Components in 3D Space

Cartesian vector notation shows how to write vectors in three-dimensional space using x, y, and z components. It gives a clear way to represent direction and size for quantities used in mechanical engineering. The notation uses the unit vectors i, j, and k for the x, y, and z axes. A vector pointing in the direction (3, −4, 5) can be written as A = 3i - 4j + 5k. The unit vectors are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1). This form is useful for describing displacement, velocity,...

direction cosines of a vector

Vector Orientation with Direction Cosines

Direction cosines describe how a vector is oriented relative to the x, y, and z axes. For a vector A written in Cartesian form with i, j, and k unit vectors, this idea helps connect the vector’s components to its direction in space. The magnitude of vector A is the square root of the sum of the squares of its components. Its direction is described by the coordinate direction angles α, β, and γ with the x, y, and z axes. These angles can be found by projecting vector A onto each axis. The...

three-dimensional force system

Force Components in 3D Space

A three-dimensional force system describes forces acting in x, y, and z directions. In mechanical engineering, this framework helps students and engineers study how objects and structures behave in three-dimensional space. One common example is a force acting on a particle in space. That force can be split into three parts: the x-component, the y-component, and the z-component. Vector mathematics is used to calculate each component, and together they make up the full three-dimensional force...

three-dimensional force system:problem solving

three-dimensional-force-systemproblem-solving

Resolving Forces in 3D Coordinates

Three-dimensional force systems involve forces acting in three different directions at the same time. These problems are common in physics and engineering. They are used to find the resultant force, which helps predict or analyze how an object or structure will behave. The first step is to resolve each force into scalar components. A scalar component is the amount of force along one axis. Trigonometric functions and vector addition are used to break the force into parts and then combine those...

position vectors

Using Position Vectors to Locate Points

Position vectors are used to locate points in space and show the direction and distance between two points. In math and science, they help describe spatial relationships and make calculations easier. A position vector can connect one point to another point or show a point’s location relative to the origin of coordinates. For a point P(x, y, z) relative to the origin O, the position vector r runs from O to P. In Cartesian vector form, it is written as r = x i + y j + z k. Here, i, j, and k are...

force vector along a line

Finding a Cable Force from Two Points

A cable force can be found from the two points that define its line of action. In three-dimensional statics, this setup often appears with a pole and a cable anchored to the ground. The cable ends are labeled A and B, and their position vectors are written as P_A and P_B. The position vector from A to B is found using the triangle law of vector addition. First, subtract P_A from P_B to get P_AB. Then find the magnitude of P_AB by taking the square root of the sum of the squares of its...

dot product

Using Dot Product to Find Work and Angle

The dot product connects vectors, work, and direction in mathematics and physics. In engineering, it is found by multiplying the magnitudes of two vectors and the cosine of the angle between them. It is shown with a dot symbol between the vectors. A real-world example is a vehicle pulling an object along the ground with a rope. If the rope is angled above the horizontal axis, the work done can be calculated with the dot product of the applied force and the object’s displacement. This makes the...

dot product: problem solving

Using the Dot Product to Solve Vector Questions

The dot product helps solve vector questions by combining vector size and the cosine of the angle between them. It is useful when the angle is measured anti-clockwise and when a problem asks for a vector relationship in component form. A good first step is to identify the question being asked. Then list the given vectors and write them in Cartesian form, also called component form. This makes the data easier to use in a calculation. Next, decide whether the dot product is the right operation.

cross product

Cross Product Direction and Magnitude

The cross product is a vector operation in vector algebra that creates a third vector from two input vectors. Unlike the scalar product, it does not give a single number. The result is a vector perpendicular to both original vectors. The size, or magnitude, of the cross product depends on the magnitudes of both vectors and the sine of the angle between them. A larger angle between the vectors gives a larger cross product magnitude. The direction is found with the right-hand rule. If the...

key terms and concepts

Learn how to break a force into x- and y-components, use trigonometry and angle theta, and understand a structure's net force in a plane.

Resolving Forces in a Plane | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Resolving Forces in a Plane

Description

Transcript