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

Mühendislik mekaniğinde cismin hareketini tanımlamak için kullanılan çeşitli fiziksel büyüklükler, skaler veya vektörler kullanılarak ölçülür.
Skalerler, büyüklükle tanımlanan ve pozitif veya negatif fiziksel bir birime sahip sayısal bir değerle temsil edilen fiziksel büyüklüklerdir. Uzunluk, hız, mesafe ve zaman bazı örneklerdir.
Vektörler, hem büyüklüğe hem de yöne sahip bir okla tanımlanan fiziksel büyüklüklerdir. Okun uzunluğu büyüklüğü temsil eder ve vektör ile sabit bir eksen arasındaki açı, hareket çizgisinin yönünü temsil eder.
Kalın harflerle veya harfin üzerinde bir ok yardımıyla temsil edilirken, büyüklükleri harfi italik hale getirerek temsil edilir. Bazı örnekler hız, ivme ve kuvvettir.
Örneğin, hareketli bir asansör düşünün. Asansörün boyutları ve kütlesi skaler büyüklüklerdir, oysa asansörün hızı ve ağırlığı hem büyüklükleri hem de yönleri olduğu için vektörel büyüklüklerdir.

<ol>
	<li>Hibbeler, R.C. (2016). Mühendislik Mekaniği \u2012 Statik ve Dinamik. Hoboken, New Jersey: Pearson Prentice Salonu. sayfa 17</li>
</ol>

skaler ve vektörel

Mekanikte, kuvvet, hız, ivme ve iş gibi yaygın olarak kullanılan terimler, skaler veya vektör miktarları olarak sınıflandırılabilir. Skaler, yalnızca büyüklüğüyle tanımlanabilen ve herhangi bir yönlendirme bileşeni gerektirmeyen bir fiziksel miktarı ifade eden bir terimdir. Skaler miktar örnekleri kütlenin, alanın ve uzunluğunun miktarıdır.

Aynı fiziksel birimlere sahip skaler miktarlar, sayılar için kullanılan normal cebir kurallarına göre eklenip çıkarılabilir. Örneğin, 50 dakikadan 10 dakika daha erken biten bir ders 50 dakika - 10 dakika = 40 dakika sürer. Benzer şekilde, 60 kalorilik bir mısır servisinden sonra 200 kalorilik bir donut servisi, enerji olarak 60 kalori + 200 kalori = 260 kalori sağlar. Bir skaler miktarı bir sayıyla çarptığımızda, aynı skaler miktarı elde ederiz, ancak daha büyük (veya daha küçük) bir değere sahip oluruz. Örneğin, dünkü kahvaltı 200 kalori enerjisi varsa ve bugünkü kahvaltı dünkünden dört kat daha fazla enerjiye sahipse, bugünkü kahvaltı 4 x 200 kalori = 800 kalori enerjiye sahiptir. İki skaler miktar da birbiriyle çarpılabilir veya bölünebilir ve türetilmiş bir skaler miktar oluşturulabilir. Örneğin, bir tren 1 saat içinde 120 km yol kat ederse, hızı 120.000 m / 3600 s = 33 m / s olur, burada hız zamana bölmeyle elde edilen türetilmiş bir skaler miktarıdır.

Öte yandan vektör miktarı, büyüklük ve yönde fiziksel bir miktarı ifade eder. Bir vektör, bir ok kullanarak grafiksel olarak temsil edilebilir. Okun uzunluğu vektörün büyüklüğünü sembolize ederken, sabit bir eksenle arasındaki açı çizgiye olan hareket hattını belirler. Okun başı vektörün işaret ettiği yönu temsil eder. Yol katetme, hız, konum, kuvvet ve tork gibi vektör miktarı örneklerine örnek verilebilir.

Bir arkadaşa bir kamp yolculuğu sırasında bir dinlenme noktasını çadırdan 6 km uzaklıkta tarif etmeyi düşünün. En kolay yol, tam konuma ulaşmak için hangi yöne gidileceğini belirlemektir; örneğin, 6 km güneydoğuya gitmek daha yardımcı olur. Bu miktar hareketin yer değiştirmesidir.

Miktarları eşit olan iki vektör, aynı büyüklükleri ve aynı yönlere sahip olduklarında eşittir. Aynı yönde olan iki vektöre paralel vektörler denir. Ancak bir vektör, birinci vektöre tam olarak 180Â° zıt yönde işaret ediyorsa, bunlara antiparalel denir. Birbirine dik yönlere sahip vektörler dik vektörler olarak adlandırılır.

Skaler ve Vektörel

Mekanikte, kuvvet, hız, hız ve iş gibi yaygın olarak kullanılan terimler skaler veya vektörel büyüklükler olarak sınıflandırılabilir. Skaler fiziksel bir ...

Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

Force Vectors

Kuvvet Vektörleri

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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...

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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,...

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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...

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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,...

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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...

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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

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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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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...

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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

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mechanical engineering

Scalar vs Vector Quantities in Mechanics

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