Name: force components in 3d space
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 30 s
Channel: JoVE

一艘船系在三维坐标系原点处的码头上。通过将绳索中的张力分解为三维空间中的分量，可以确定该张力的大小。
考虑包含力矢量并穿过垂直 z 轴的平面 OLMN。
该力与z轴成角度：theta，而平面OLMN与x轴成角度phi。
平面 OLMN 中力的垂直分量和水平分量可以表示为角度 theta 的三角函数。
然而，该水平分量可进一步分解，得到沿“x”轴和“y”轴的标量分量，这些分量可用角度 phi 表示。
代入水平分量后，可将 x 和 y 分量表示为角度 theta 和 phi 的函数。
力 Fh 的大小可通过直角三角形 ONP 中的勾股定理求得。 类似地，在三角形 OLM 中应用勾股定理，可将力 F 的大小表示为其在 x、y 和 z 轴方向上的分量形式。

<ol>
	<li>希伯勒，RC（2016 年）。工程力学 \u2012 静力学和动力学。 新泽西州霍博肯:Pearson Prentice Hall。第 47-49 页。</li>
</ol>

三维力系统

Three-Dimensional Force System

在机械工程中，三维力的系统是一种在三个维度中产生作用力的系统，它们分别沿着x轴、y轴和z轴来施加力。三维力的系统是机械工程中的重要概念，它使工程师能够理解和分析物体和结构在三维空间中的行为。通过理解作用在系统上的力，工程师可以设计更高效和有效的机械系统，使其能够承受外力并在不同的条件下运行。 一个三维力系统的例子是作用在空间粒子上的力。这个力可以分解为三个分量：x分量、y分量和z分量。每个分量都可以通过向量的值来进行计算，它们共同构成了一个三维力系统。 另一个三维力系统的例子是作用在结构上的力，比如桥梁或建筑物。在这种情况下，力是分布在整个结构中的，它们既可以是外力，比如来自风和地震的力，也可以是内力，比如由结构自身重量所引起的力。 理解三维力系统的行为对于设计和分析机械系统至关重要。工程师通过使用各种技术来分析这些系统，包括向量值、静力学和动力学。通过分析作用在系统上的力，工程师可以确定在不同条件下它的行为，并且以此来判断出潜在的失效点。

在机械工程中，三维力系是指在三维空间中作用的力系统，其力分别沿 x、y 和 z 方向施加……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Force Vectors

力矢量

KEY TERMS AND CONCEPTS

关键术语与概念

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 system.

A second example is the forces on a structure such as a bridge or a building. These forces can spread through the structure. They may include external forces like wind and earthquakes, as well as internal forces from the structure’s own weight.

Understanding these forces matters in design and analysis. Engineers use vector mathematics, statics, and dynamics to study how a system will respond under different conditions. This analysis can also help them find possible points of failure before a mechanical system is built or used.

Force Components in 3D Space

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

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introduction to force

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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...

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Resultant Force from Tugboats

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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

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

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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

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

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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

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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 forces in x, y, and z directions are split into components and analyzed in particles and structures like bridges and buildings.

Force Components in 3D Space | 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

Force Components in 3D Space

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