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

Consider a cyclist pedaling a bicycle. When a force is exerted on the pedal, the crankshaft rotates such that the chain attached to the crankshaft exerts a force on the cogs of the wheel.
On establishing a coordinate system, the tangential force on the cogs is along the x-axis, while the radius of the cogs is the moment arm expressed in terms of the position vector.
Here, the moment of force about point O is along the y-direction perpendicular to both the force and position vector and is expressed as the cross product of the position and force vectors.
The cross-product can be solved by expressing the coefficient of components of force and position vector in the form of a determinant.
Expanding the determinant, the moment&#39;s component along the x-axis is expressed in terms of the y and z components of the force and position vector.
Similarly, expressing its other two components in terms of force and position vector components, the moment of a force can be expressed in cartesian vector form.

moment of a force about an axis: vector

When a force is exerted on an object, it can cause that object to rotate about an axis. The moment of a force, also known as torque, measures the force&#39;s ability to cause that rotation. In the case of a cyclist pedaling a bicycle, the force exerted on the pedal causes the crankshaft to rotate, which in turn causes the wheel to spin. The moment of the force exerted on the pedal drives the wheel&#39;s rotation.
First, establish a coordinate system to understand how the moment of a force works. Consider the bicycle as an example. Establish the x-axis to be tangential to the wheel, and the y-axis to be perpendicular to the x-axis and pointing upwards. The z-axis can be pointing outwards from the wheel.
When the cyclist exerts a force on the pedal, the crankshaft rotates such that the chain attached to the crankshaft exerts a force on the axle of the wheel. The force exerted on the axle is then along the x-axis. The radius of the axle is the moment arm, which can be expressed in terms of the position vector.
The moment of force about point O is along the z-direction, perpendicular to both the force and position vector. It is expressed as the cross-product of the position and force vectors.
The cross-product can be solved by expressing the coefficient of components of force and position vector in the form of a determinant. The determinate can be expanded to solve for each component of the moment of force.
For example, the moment&#39;s component along the x-axis is expressed in terms of the y and z components of the force and position vector. Similarly, the other two components of the moment can be expressed in cartesian vector form.
Understanding the moment of force is important in many fields of science and engineering. It can help us calculate the forces on objects such as gears, pulleys, and levers and can aid in designing machines that convert forces into movement. In the bicycle example, understanding the moment of force can help us optimize the design of the crankshaft and chain to maximize the force exerted on the wheel.

Moment of a Force About an Axis: Vector

When a force is exerted on an object, it can cause that object to rotate about an axis. The moment of a force, also known as torque, measures the ...

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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...

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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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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...

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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).

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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...

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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...

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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simplification of a force and couple system i

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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

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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...

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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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