Name: torque on a bicycle wheel: vector method
Uploaded: 2023-09-22T11:49:21.000Z
Duration: 1 min 29 s
Channel: JoVE

Wyobraźmy sobie rowerzystę pedałującego na rowerze. Gdy na pedał wywierana jest siła, wał korbowy obraca się tak, że łańcuch przymocowany do wału korbowego wywiera siłę na zębatki koła.
Po ustaleniu układu współrzędnych siła styczna działająca na koła zębate przebiega wzdłuż osi x, podczas gdy promień kół zębatych jest ramieniem momentu wyrażonym w wektorze położenia.
W tym przypadku moment siły wokół punktu O przebiega wzdłuż kierunku y prostopadłego zarówno do wektora siły, jak i położenia i jest wyrażony jako iloczyn krzyżowy wektorów położenia i siły.
Iloczyn krzyżowy można rozwiązać, wyrażając współczynnik składowych wektora siły i położenia w postaci wyznacznika.
Rozszerzając wyznacznik, składowa momentu wzdłuż osi x jest wyrażona jako składowe y i z wektora siły i położenia.
Podobnie, wyrażając jego pozostałe dwie składowe w kategoriach składowych wektora siły i położenia, moment siły można wyrazić w kartezjańskiej postaci wektorowej.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska \u2012 Statyka i dynamika. Hoboken, New Jersey: Pearson Prentice Hall. Strona 146</li>
	<li>Piwo, F.P.; Johnston, E.R.; Mazurek, D.F.; Cromwell, PJ i Jaźń, BP (2019). Mechanika wektorowa dla inżynierów \u2012 Statyka i dynamika. Nowy Jork: McGraw-Hill. Strony 107 - 109</li>
</ol>

moment siły wokół osi: wektor

Kiedy siła działa na przedmiot, może wywołać obrót tego przedmiotu wokół osi. Moment siły, znany również jako moment obrotowy, mierzy zdolność siły do wywołania takiego obrotu. W przypadku jadącego na rowerze rowerzysty, siła wywierana na pedał powoduje obrót korbowodu, co z kolei prowadzi do obrotu koła. Moment siły wywieranej na pedał napędza obrót koła.

Na początek ustalmy układ współrzędnych, aby zrozumieć, jak działa moment siły. Weźmy rower jako przykład. Ustalmy, że oś x będzie styczna do koła, a oś y będzie prostopadła do osi x i skierowana w górę. Oś z może być skierowana na zewnątrz od koła.

Kiedy rowerzysta wywiera siłę na pedał, korbowód obraca się tak, że łańcuch przyczepiony do korbowodu wywiera siłę na osi koła. Siła wywierana na oś jest wtedy wzdłuż osi x. Promień osi stanowi ramię momentu, które można wyrazić za pomocą wektora położenia.

Moment siły wokół punktu O działa w kierunku osi z, prostopadły zarówno do siły, jak i wektora położenia. Wyraża się go jako iloczyn wektorowy wektora położenia i wektora siły.

Iloczyn wektorowy można obliczyć, wyrażając współczynniki składowe wektora siły i wektora położenia w postaci wyznacznika. Wyznacznik można rozwinąć, aby rozwiązać dla każdej składowej momentu siły.

Na przykład, składowa momentu wzdłuż osi x jest wyrażona za pomocą składowych y i z siły oraz wektora położenia. Podobnie pozostałe dwie składowe momentu można wyrazić w postaci wektora kartezjańskiego.



Zrozumienie momentu siły jest kluczowe w wielu dziedzinach nauki i inżynierii. Może pomóc w obliczeniu siły na przedmioty, takie jak koła zębate, koła pasowe i dźwignie, oraz może pomóc w projektowaniu maszyn, które zamieniają siły w ruch. W przykładzie roweru zrozumienie momentu siły może pomóc w zoptymalizowaniu projektu korbowodu i łańcucha w celu maksymalizacji siły wywieranej na koło.

Moment siły wokół osi: Wektor

Kiedy siła jest wywierana na obiekt, może to spowodować, że obiekt ten obróci się wokół osi. Moment siły, znany również jako moment obrotowy, mierzy ...

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Force System Resultants

wypadkowe układu sił

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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 radius acts as the moment arm, and it can be written using the position vector. The moment of the force about point O points in the z-direction because it is perpendicular to both the force and the position vector. It is found using the cross product of the position and force vectors.

The cross product can be calculated by writing the force and position components in determinant form. Expanding the determinant gives the x-, y-, and z-components of the moment. The x-component, for example, depends on the y and z components of the force and position vectors.

Knowing how to find the moment of a force is useful in science and engineering. It helps with problems involving gears, pulleys, and levers. It also supports the design of machines that turn force into motion, including the bicycle crankshaft and chain system.

Torque on a Bicycle Wheel: Vector Method

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...

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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

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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

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, 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

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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...

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Couple Moments and the Right-Hand Rule

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Equivalent Couples in Mechanical Systems

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moment of a couple: problem solving

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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

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Resultant Forces in Force and Couple Systems

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simplification of a force and couple system: ii

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

Resultant Force and Wrench Motion

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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...

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key terms and concepts

Learn how torque, or the moment of a force, makes a bicycle wheel rotate. Follow the vector method, coordinate axes, and cross product step by step.

Torque on a Bicycle Wheel: Vector Method | JoVE K12

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mechanical engineering

Torque on a Bicycle Wheel: Vector Method

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