Name: scalar vs vector quantities in mechanics
Uploaded: 2023-09-22T11:49:38.000Z
Duration: 1 min 22 s
Channel: JoVE

Várias grandezas físicas na mecânica de engenharia usadas para descrever o movimento do objeto são medidas usando escalares ou vetores.
Escalares são grandezas físicas descritas por magnitude e representadas por um valor numérico com uma unidade física, positiva ou negativa. Comprimento, velocidade, distância e tempo são alguns exemplos.
Vetores são quantidades físicas descritas por uma seta com magnitude e direção. O comprimento da seta representa a magnitude, e o ângulo entre o vetor e um eixo fixo representa a direção de sua linha de ação.
Eles são representados em negrito ou com a ajuda de uma seta sobre a letra, enquanto suas magnitudes são representadas em itálico na letra. Alguns exemplos são velocidade, aceleração e força.
Por exemplo, considere um elevador em movimento. As dimensões e a massa do elevador são as grandezas escalares, enquanto a velocidade e o peso do elevador são grandezas vetoriais, pois têm uma magnitude e uma direção.

<ol>
	<li>Hibbeler, R.C. (2016). Engenharia Mecânica \u2012 Estática e Dinâmica. Hoboken, Nova Jersey: Pearson Prentice Hall. pág. 17</li>
</ol>

escalares e vetores

Em mecânica, termos comumente usados como força, velocidade, velocidade e trabalho podem ser classificados como quantidades escalares ou vetoriais.

Um escalar é uma quantidade física que pode ser descrita apenas por sua magnitude e não requer componentes direcionais. Exemplos de quantidades escalares são massa, área e comprimento. Quantidades escalares com as mesmas unidades físicas podem ser adicionadas ou subtraídas de acordo com as regras usuais da álgebra para números. Por exemplo, uma aula que termina 10 minutos mais cedo do que 50 minutos dura 50 minutos − 10 minutos = 40 minutos. Da mesma forma, uma porção de 60 calorias de milho seguida por uma porção de 200 calorias de rosquinhas dá 60 calorias + 200 calorias = 260 calorias de energia. Quando multiplicamos uma quantidade escalar por um número, obtemos a mesma quantidade escalar, mas com um valor maior (ou menor). Por exemplo, se o café da manhã de ontem teve 200 calorias de energia e o café da manhã de hoje tem quatro vezes mais energia do que ontem, então o café da manhã de hoje tem 4 × 200 calorias = 800 calorias de energia. Duas quantidades escalares também podem ser multiplicadas ou divididas uma pela outra para formar uma quantidade escalar derivada.Por exemplo, se um trem percorre uma distância de 120 km em 1h, sua velocidade é 120.000 m / 3600 s = 33 m / s, onde a velocidade é uma quantidade escalar derivada obtida pela divisão da distância pelo tempo. Por outro lado, uma quantidade vetorial é uma quantidade física que possui magnitude e direção.

Um vetor pode ser representado graficamente usando uma seta. O comprimento da seta simboliza a magnitude do vetor, enquanto o ângulo entre ela e um eixo fixo determina sua linha de ação.

A ponta da seta representa a direção na qual o vetor está apontando. Exemplos de quantidades vetoriais incluem deslocamento, velocidade, posição, força e torque. Considere dar instruções a um amigo em uma viagem de caminhada para um ponto de descanso a 6 km da barraca. A maneira mais fácil será identificar em qual direção chegar ao local exato; por exemplo, 6 km a sudeste será mais útil. Essa quantidade é o deslocamento do caminhante.

Dois vetores são iguais se tiverem magnitudes iguais e a mesma direção. Dois vetores com direções idênticas são ditos vetores paralelos. No entanto, se um vetor aponta na direção oposta, exatamente 180°, ao primeiro vetor, eles são ditos antiparalelos. Dois vetores com direções perpendiculares entre si são ditos vetores ortogonais.

Escalares e Vetores

Em mecânica, termos comumente usados como força, velocidade, velocidade e trabalho podem ser classificados como quantidades escalares ou vetoriais. Um escalar é um físico ...

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

Force Vectors

Vetores de Força

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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 min.

Scalars can also be multiplied by a number. If yesterday’s breakfast had 200 calories of energy and today’s breakfast has four times as much, then today’s breakfast has 800 calories of energy. Two scalars can also be multiplied or divided to make another scalar quantity. For example, a train that covers 120 km in 1 h has a speed of 120,000 m/3600 s, or 33 m/s. That speed is a derived scalar quantity.

A vector has both magnitude and direction. Vectors are often shown with arrows. The arrow length shows the magnitude. The angle with a fixed axis shows the line of action, and the arrowhead shows the direction. Displacement, velocity, position, force, and torque are examples of vector quantities.

Direction matters in everyday situations. If a hiking rest point is 6 km from a tent, saying 6 km southeast is more useful than saying only 6 km. That gives the hiker displacement, which is a vector. Two vectors are equal if they have the same magnitude and the same direction. Vectors with the same direction are parallel. Vectors that point exactly 180° apart are antiparallel. Vectors that meet at right angles are orthogonal.

Scalar vs Vector Quantities in Mechanics

scalar and vectors

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...

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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

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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

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two-dimensional force system: problem solving

two-dimensional-force-system-problem-solving

Two-Dimensional Forces: Free-Body Analysis

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scalar notation

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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

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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

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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...

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Finding a Cable Force from Two 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 scalar and vector quantities differ in mechanics, with clear examples of magnitude, direction, displacement, speed, and force for high school science.

Scalar vs Vector Quantities in Mechanics | JoVE K12

O JoVE Core Mechanical Engineering oferece vídeos animados concisos e de fácil compreensão que abrangem tópicos-chave como equilíbrio, forças, análise de treliças e mecânica estrutural. Explore suas aplicações no mundo real e observe esses conceitos em ação por meio de experimentos conduzidos por cientistas.

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Scalar vs Vector Quantities in Mechanics

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