Name: vector components for force calculations
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 28 s
Channel: JoVE
Description: 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.

한 남자가 북동쪽 방향의 갈고리에서 밧줄을 당기고 있다고 생각해 보십시오.
힘 벡터는 F1로 표시됩니다. x축을 따라 F1x, y축을 따라 F1y로 표시되는 스칼라 구성 요소로 해석됩니다.
시계 반대 방향으로 양의 x축을 가진 F1 에 의해 만들어진 각도 인 방향은 x 구성 요소에 대한 y 구성 요소의 황갈색 역입니다.
여기서, F1x와 F1y는 직각 삼각형을 형성하는 직사각형 성분으로서, 삼각함수를 이용하여 구할 수 있다.
다른 힘 F2 가 남동쪽 방향에서 후크에 작용하면 F2x 는 양의 x 축을 따라 있고 F2y 는 음의 y 축을 따라 있습니다.
F2 의 x 및 y 성분은 삼각 함수를 사용하여 구하며, 결과 힘은 x 축과 y 축을 따라 두 힘의 성분의 대수 적 합이며, 그 크기는 구성 요소의 제곱 합의 제곱근을 사용하여 얻습니다.

<ol>
	<li>맥주, F.P.; 존스턴, ER; 마주렉, D.F; 크롬웰, PJ 및 셀프, BP (2019). 엔지니어를 위한 벡터 역학 \u2012 Statics and Dynamics. 뉴욕 : McGraw-Hill. 29-33쪽</li>
	<li>히벨러, R.C. (2016). Engineering Mechanics \u2012 정역학 및 동역학. 뉴저지 호보켄: 피어슨 프렌티스 홀. 33-35쪽.</li>
</ol>

스칼라 표기법

스칼라 표기법은 벡터와 관련된 계산을 단순화하는 데 유용한 방법입니다. 벡터를 더하거나 뺄 때 스칼라 표기법을 사용하여 벡터의 성분을 별도로 더하거나 뺄 수 있습니다. 예를 들어 벡터 양인 힘은 직사각형 성분이라고 하는 x 및 y 성분으로 분해될 수 있으며, 그런 다음 삼각함수를 사용하여 이러한 성분의 크기와 방향을 결정할 수 있습니다. 

북동쪽 방향으로 갈고리에서 밧줄을 당기는 남자를 생각해 봅니다. 이 적용된 힘 벡터의 크기는 F1으로 표시됩니다. 이는 x축을 따라 F1x으로, y축을 따라 F1y로 표시되는 스칼라 성분으로 분석됩니다. 직사각형 성분 F1x과 F1y은 직각 삼각형을 이루기 때문에 삼각함수를 이용하여 구할 수 있습니다. 이 성분들과 피타고라스 정리를 이용하여 힘 F1의 크기를 계산할 수 있습니다. x성분 위에 있는 y성분의 탄젠트 역은 힘의 방향을 알려줍니다. 남동쪽 방향에서 또 다른 힘 2가 동일한 고리에 작용하면, 비슷한 방법으로 이 힘의 크기와 방향도 찾을 수 있습니다.

합력은 x 및 y 축을 따라 작용하는 두 힘의 성분을 대수적으로 합한 것입니다. 그 크기는 성분의 제곱합의 제곱근을 사용하여 얻을 수도 있습니다. 이 합력은 물체에 가해지는 총 힘을 나타낼 수도 있고 다른 힘에 대응하는 데 필요한 힘을 나타낼 수도 있습니다.

스칼라 표기법은 다양한 방향의 힘을 계산하고, 물체에 작용하는 힘을 이해하는 데 유용합니다. 힘을 직사각형 성분으로 나눈 다음 삼각 함수를 사용하면 힘의 크기와 방향을 빠르고 정확하게 결정할 수 있습니다.

스칼라 표기법은 벡터와 관련된 계산을 단순화하는 데 유용한 방법입니다. 벡터를 더하거나 뺄 때, 해당 구성 요소를 더하거나 ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Force Vectors

힘 벡터

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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.

Vector Components for Force Calculations

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

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

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 scalar notation breaks forces into x and y components so students can calculate magnitude, direction, and resultant force with trigonometry.

Vector Components for Force Calculations | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Vector Components for Force Calculations

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