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

考虑一名骑手在骑自行车。当对踏板施加一个力时，曲柄轴随之旋转，从而使连接在曲柄轴上的链条对车轮的链轮施加一个力。
建立坐标系后，齿轮上的切向力沿 x 轴方向，而齿轮的半径即为用位置矢量表示的力臂。
此处，关于点 O 的力矩方向沿 y 轴，垂直于力和位置矢量，可表示为位置矢量与力矢量的叉积。
可以通过将力和位置矢量的分量系数表示为行列式的形式来求解叉积。
展开行列式，力矩沿该方向的分量 x-轴以...表示 y 和 z 力和位置矢量的分量
同样，将其另外两个分量用力和位置矢量的分量表示，力矩可以用笛卡尔矢量形式表达。

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	<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 146 页</li>
	<li>比尔，F.P.;约翰斯顿，ER;马祖克，DF;克伦威尔，PJ 和塞尔夫，BP（2019 年）。工程师的矢量力学 \u2012 静力学和动力学。纽约:麦格劳-希尔。页码 107 - 109</li>
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关于轴的力矩：矢量

Moment of a Force About an Axis: Vector

当一个力作用在物体上时，它能够使得物体围绕一个轴来进行旋转。力矩，也被称为扭矩，衡量了由力来引起旋转的能力。以骑自行车为例，踩踏板所施加的力使得曲柄轴发生旋转，进而使得车轮发生旋转。施加在踏板上的力矩驱动了车轮的旋转。 首先，通过建立一个坐标系来理解力矩的工作原理。以自行车为例，将x轴与车轮切线方向平行，将y轴与x轴垂直并使其向上。z轴则可以指向车轮的外部。 当骑自行车的人对踏板施加力时，曲柄轴便会发生旋转，使得连接曲柄轴的链条施加的力位于车轮的轴上。施加在轴上的力将会沿着x轴的方向。其中轴的半径是力臂，可以使用位置向量来进行表示。 O点的力矩是沿着z轴的方向，并且垂直于力和位置向量。它可以表示为位置向量与力向量的叉乘。 通过将力和位置向量的分量系数表示为行列式的形式，可以通过叉乘来进行求解。可以将行列式进行展开来对力矩的每个分量进行求解。 举例来说，力矩在x轴方向上的分量可以用力和位置向量上的y和z分量来对其进行表示。同样的，力矩的其他两个分量可以用笛卡尔矢量的形式来对其进行表示。 力矩在许多科学和工程领域中是非常重要的。它可以帮助我们来计算齿轮、滑轮和杠杆等物体的受力情况，并且可以通过辅助设计将力转化为运动的机器。在自行车的例子中，通过理解力矩可以帮助我们优化对曲柄轴和链条的设计，以最大的限度地来增加在车轮上所施加的力。

当一个力作用于物体时，可使该物体绕轴发生旋转。力矩，也称为转矩，用于度量……

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KEY TERMS AND CONCEPTS

关键术语与概念

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...

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...

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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

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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Couple Moments and the Right-Hand Rule

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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...

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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 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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Torque on a Bicycle Wheel: Vector Method

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