Name: calculating moment with cartesian vectors
Uploaded: 2025-05-22T09:06:02.000Z
Duration: 1 min 26 s
Channel: JoVE

מומנט הכוח מוגדר כמכפלה צולבת של וקטורי מיקום וכוח.
המכפלה הצולבת כרגע יכולה לבוא לידי ביטוי בצורה הדטרמיננטית, באמצעות וקטורי יחידה והצורה הקרטזית של וקטורי מיקום וכוח.
הדטרמיננטה מורחבת כדי לקבוע את ניסוח הווקטור הקרטזי של רגע הכוח.
קחו בחשבון שכוח של 10 N מופעל על דלת מסתובבת בנקודה הנתונה על ידי וקטור המיקום בצורת הרכיבים הקרטזיים. כאן יש לקבוע את רגע הכוח בניסוח הווקטורי.
במקרה זה, וקטור הכוח ניצב למישור y-z , וכתוצאה מכך רכיבי y ו- z של וקטור הכוח הם אפס. דטרמיננטה נבנית באמצעות וקטור המיקום ווקטור הכוח.
לבסוף, על ידי הרחבת הדטרמיננטה, נקבע רגע הכוח של הדלת המסתובבת על ציר z.
כאן, הרגע הוא בכיוון z חיובי, מה שמרמז על כך שהדלת מסתובבת נגד כיוון השעון.

<ol>
	<li>היבלר, R.C. (2016). מכניקה הנדסית - סטטיקה ודינמיקה. הובוקן, ניו ג'רזי: אולם פירסון פרנטיס. עמ' 129.</li>
</ol>

צורה קרטזיאנית לניסוח וקטור

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 הצורה הקרטזיאנית לניסוח וקטור היא תהליך לחישוב רגע המומנט באמצעות וקטורי המיקום והכוח. מומנט הכוח מוגדר כמכפלה וקטורית של וקטורים אלה, מה שהופך אותו לכמות וקטורית. הצורה הקרטזיאנית של וקטורי המיקום והכוח כוללת וקטורי יחידה, שבהם ניתן להשתמש כדי לבטא את המכפלה הצולבת בצורה דטרמיננטית. 
 <img alt="Equation 1" src="/files/ftp_upload/14254/14254_ Equation_1v2.svg" style="text-align: center; height: 60px;"> 


 ניתן להרחיב את הקובע כדי לקבוע את הניסוח הווקטור הקרטזיאני של מומנט הכוח. 
 <img alt="Equation 2" src="/files/ftp_upload/14254/14254_ Equation_2v2.svg" style="text-align: center; height: 15px;"> 

 לדוגמה, נניח שכוח של 10 N מופעל על דלת מסתובבת לאורך כיוון x בנקודה הניתנת על ידי וקטור המיקום r = 0i - 4j + 0k בצורת הרכיב הקרטזיאני. במקרה זה, וקטור הכוח מאונך למישור y-z שלאורכו עומדת הדלת, מה שאומר שהרכיבים y ו-z של וקטור הכוח הם אפס. כדי לחשב את מומנט הכוח, נבנה דטרמיננט באמצעות וקטור המיקום ווקטור הכוח, F = 10i + 0j + 0 k.
 <img alt="Equation 3" src="/files/ftp_upload/14254/14254_ Equation_3v2.svg" style="text-align: center; height: 60px;"> 

 לבסוף, על ידי הרחבת הקובע, נוכל לקבוע שמומנט הכוח של הדלת המסתובבת שווה ל-40 ננומטר סביב ציר z. המומנט החיובי מרמז שהדלת מסתובבת נגד כיוון השעון. 

 הצורה הקרטזיאנית לניסוח וקטור מאפשרת לנו לחשב את מומנט הכוח באמצעות וקטורי מיקום וכוח בצורה פשוטה.

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הצורה הקרטזית לניסוח וקטורי היא תהליך לחישוב רגע הכוח באמצעות וקטורי המיקום והכוח. רגע הכוח הוא...

Education

JoVE Core

Mechanical Engineering

הנדסת מכונות

Unit 1

יחידה 1

Force System Resultants

תוצאות מערכת כוחות

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

Cartesian Form for Vector Formulation

calculating moment with cartesian vectors

Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form.

The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step.

A revolving door example shows the process in action. A force of 10 N is applied in the x-direction at the position vector r = 0 i - 4 j + 0 k. The force vector has no y or z components, so it can be written as F = 10 i + 0 j + 0 k.

Using these vectors, the determinant is built and expanded. The result is a moment of 40 N.m about the z-axis. The positive sign means the door rotates counterclockwise.

Calculating Moment with Cartesian Vectors

calculating-moment-with-cartesian-vectors

moment of a force: scalar formulation

Torque on a Flywheel: Direction and Size

Torque, or the moment of a force, describes how a force can make an object turn about an axis. It is a vector quantity, so it has both size and direction. This idea is important in physics, mechanics, and engineering. A simple flywheel example shows how torque works. The flywheel rotates about point O when a force is applied to it. The moment arm is the perpendicular distance from point O to the line of action of the force. The moment, in newton-meter (N-m), is found by multiplying the force...

moment of a force: problem solving

Calculating Force Moments in Problem Solving

Calculating force moments in problem solving helps students analyze rotation in mechanical systems. The scalar form of the moment of a force is used to work through these problems step by step. Start by drawing a free-body diagram, or FBD. An FBD is a sketch that shows every force acting on the system. Each force must be identified and included. Next, identify the axis or point about which the moment is calculated. In some problems, that axis or point is given. In other cases, you must choose...

resultant moment: scalar formulation

Net Moment Sign Convention in 2D Forces

Net moment in two-dimensional force systems shows how multiple forces can cause rotation about a fixed point. The scalar formulation helps analyze equilibrium in structures that are acted on by several forces. It focuses on the x-y plane and uses a simple sign rule to describe the direction of rotation. To find the resultant moment, the moments from all forces in the system are considered about a chosen point A. Each moment is assigned a sign based on its sense of rotation. Positive moments...

moment of a force: vector formulation

Moment of Force and Direction Rules

The moment of force describes the twisting effect a force can have on an object. It shows why a force may cause rotation instead of straight-line motion. The moment arm is the perpendicular distance from the force’s line of action to the axis of rotation. The moment of force is a vector quantity, not a scalar. Its vector form comes from the cross product of the position vector and the force vector. A cross product gives a vector that is perpendicular to both input vectors. The moment about a...

cartesian form for vector formulation

Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form. The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step. A revolving door...

resultant moment: vector formulation

Calculating Torque from Multiple Forces

Resultant moment describes the total turning effect of multiple forces acting on an object. It shows how several forces can combine to make a body rotate about a point. In physics, moment is the tendency of a force to cause rotation. The resultant moment of a force system is a vector quantity. That means it can be added and subtracted like other vectors. For a system with two forces, F1 and F2, acting on a pole at points A and B, the resultant moment is written as MR = (rA x F1) + (rB x F2).

principle of moments

Resolving Forces in Moment Calculations

The principle of moments, also called Varignon's theorem, explains how a force creates rotation about a point. It is a key idea in physics and engineering. It also helps describe when a rigid body stays in equilibrium under external forces. The moment of a force is found by multiplying the force by the perpendicular distance from the point of application to the point being studied. A larger force or a larger distance gives a larger moment. This makes the idea useful for checking how strongly a...

principle of moments: problem solving

Torque About an Axis in 3D Systems

The principle of moments helps solve torque problems in physics and engineering. It describes how forces and moments balance around a pivot point or axis. It is used in many real-life cases, including construction, sports, and everyday actions like opening doors and pushing objects. The problem here uses a pole in a three-dimensional system with a cable attached. When tension acts in the cable, the goal is to find the moment about the z-axis through the base. One method is to project the force...

moment of a force about an axis: scalar

Bicycle Torque and Moment About an Axis

The moment of a force about an axis describes how a force can make an object turn around a fixed line. In scalar analysis, the moment is found from the perpendicular distance between the axis of rotation and the line of action of the force. That distance is called the moment arm. A bicycle gives a clear example of this idea. When the cyclist pushes on the pedal, the chain applies a force to the wheel’s axle and makes it rotate. The axis of rotation passes through the axle, and the moment arm...

moment of a force about an axis: vector

Torque on a Bicycle Wheel: Vector Method

Torque, or the moment of a force, explains how a force can make an object rotate about an axis. A bicycle is a clear example. When a cyclist presses on the pedal, the crankshaft turns and helps drive the wheel. To analyze this motion, a coordinate system is set up first. The x-axis is tangential to the wheel, the y-axis is perpendicular to the x-axis and points upward, and the z-axis points outward from the wheel. In this setup, the force on the wheel axle lies along the x-axis. The axle...

couple

Couple Moment in Force Systems

A couple is a pair of parallel forces that are equal in magnitude and opposite in direction. The two forces are separated by a perpendicular distance called the couple arm. Together, they create a turning effect on a body instead of a net push. This turning effect is called the couple moment. It rotates the body about an axis that is perpendicular to the plane of the forces. The SI unit for a couple moment is the Newton-meter (N-m). A common example is tightening a bolt with a lug wrench. In...

couples: scalar and vector formulation

Couple Moments and the Right-Hand Rule

Couple moments describe the turning effect of two parallel forces that are equal in magnitude and opposite in sense. This force pair creates rotation instead of straight-line motion. A common example is a ship’s steering wheel, where a small turn can guide the ship through the ocean. A couple moment can rotate clockwise or anticlockwise. The right-hand rule helps identify its direction. To use it, curl the fingers of your right hand in the direction of rotation. Your thumb points in the...

equivalent couples

Equivalent Couples in Mechanical Systems

Equivalent couples in mechanical systems describe pairs of forces that create the same rotational effect on a rigid body. Two couples are equivalent when they have the same moment magnitude and act in the same direction. They then cause the same angular displacement or angular acceleration in the body. A couple is made of equal and opposite forces separated by a perpendicular distance. The moment of the couple depends on both the force size and the spacing between the forces. For example, one...

moment of a couple: problem solving

Torque from a Couple: Solving Forces

The moment of a couple is a key physics and engineering idea for finding torque, or rotational force. A couple is made of two equal and opposite forces that act on an object. When those forces work together, they create a twisting effect that can rotate the object. The moment of a couple is calculated by multiplying the size of one force by the perpendicular distance between the two lines of action. That distance is the shortest straight-line distance between the force directions. This gives...

system of forces and couples

Equivalent Force and Moment Systems

Structural systems often experience several forces and couple moments at the same time. Engineers simplify these loading cases by replacing them with an equivalent force and moment system at a chosen point O. This makes the analysis easier while keeping the same external effects on the member. A key idea in this process is the principle of transmissibility. A force is a sliding vector, which means it can be moved anywhere along its line of action without changing how the body behaves...

simplification of a force and couple system i

simplification-of-a-force-and-couple-system-i

Resultant Forces in Force and Couple Systems

Force and couple systems can be reduced to a simpler equivalent system. This makes rigid body analysis easier because the external effect of the forces is captured with fewer elements. When the resultant force and resultant couple moment have perpendicular lines of action, the system may be reduced further to a single resultant force along a new line of action. Three common force systems can be simplified this way: concurrent, coplanar, and parallel. A concurrent force system has lines of...

simplification of a force and couple system: ii

simplification-of-a-force-and-couple-system-ii

Resultant Force and Wrench Motion

Resultant force and resultant couple moment describe how several forces and moments act together on a rigid body in three dimensions. When many forces act on an object, they can be replaced by one equivalent force. The moments from those forces can also be combined into one equivalent couple moment. In some cases, the resultant force and the resultant couple moment are not mutually perpendicular. That means the angle between them is not 90 degrees. To study this situation, the couple moment is...

distributed loads

Load Distribution on Shelves and Dams

Load distribution helps engineers describe how force spreads across a surface or structure. A distributed load is a continuous force per unit area, rather than a single point force. It is a common idea in many practical situations. A bookshelf full of stacked books is one simple example. The weight of the books is spread along the shelf, so the pressure changes across the surface. This pressure is measured in Newtons per square meter, or Pascals. A hydroelectric dam gives another clear...

load along a single axis

Finding Resultant Loads on a Beam

Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted. The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the...

distributed loads: problem solving

Equivalent Resultant Loads on Beams

Equivalent resultant loads on beams can be found by breaking a distributed load into smaller regions. This makes the beam easier to analyze. Each region can then be handled on its own before the results are combined. The magnitude of the equivalent resultant load for each region comes from the area of that region. In this setting, area represents the force applied by the distributed load over that part of the beam. After finding the force for each region, the position of each resultant load...

key terms and concepts

Learn how Cartesian vector form is used to calculate moment of force with unit vectors, determinants, and a revolving door example in a simple step-by-step method.

Calculating Moment with Cartesian Vectors | 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

Calculating Moment with Cartesian Vectors

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