What is the exact difference between a moment and a couple? In some YouTube channels and books, they say the moment of a force produces a translational as well as rotational motion whereas the concept of couple produces a pure rotation. How and why is it so? I am not convinced about this yet. Once you fix your point of axis or rotation axis. How come when you apply a force the body would make a translational motion?
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2 Answers
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A common way to think of forces in classical mechanics is as having a line of action as well as a direction and magnitude. (They are therefore not actually vectors.) If you push on an object at a single point, the line of action of the force goes through that point. You can add forces, and the resultant usually is a new non-zero force with magnitude, direction, and line of action. The moment measures the distance from the chosen origin to the line of action of the resultant force.
However, it can happen that the force vectors (the magnitude and direction of the forces) cancel out. The force vector is zero, and has no direction. You cannot define a line of action for the force, and therefore you cannot define a moment, which has to be the distance from the origin to the object times its magnitude. However, the 'turning tendency' of the forces does not cancel out. If the two forces are equal and opposite but applied on parallel lines of action, the object still tends to turn, but the force vector is zero, and the moment concept makes no sense.
To avoid possible confusion, the turning tendency is given a special name in this case, which is a couple. A couple has no line of action, so the 'moment' concept is not appropriate. We could redefine 'moment' or 'torque' to include this special case, but that makes the definition of a moment somewhat messy.
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There is a way to give a simple, uniform definition that covers all cases, and as a bonus unifies all the linear/angular quantities in physics, but it requires slightly more sophisticated mathematics, so is not usually taught to beginners. (And if you are happy with the above, feel free to skip the rest of this explanation!) In this approach, we have to add an extra dimension, and put the origin a unit distance 'outside the universe'. Then a force can be described as a new sort of entity called a 'bivector' - you can think of it as defining a directed plane through the origin, in the same way a vector defines a directed line through the origin.
For 2D mechanics in the $xy$ plane, we add the $w$ direction at right angles to the plane, and the real world is the $w=1$ plane offset from the origin. Then the force bivector $f$ has three components: $f_{wx}$, $f_{wy}$, and $f_{xy}$, representing the three coordinate planes. The $f_{wx}$ and $f_{wy}$ components are the two components of the linear force vector, and the $f_{xy}$ component is the torque. The plane through the origin usually cuts the $w=1$ real world plane along a line, which is the line of action of the force. The distance from the real world origin in the $w=1$ plane to the line of action is $f_{xy}/\sqrt{f_{wx}^2+f_{wy}^2}$ and we multiply this by the magnitude of the force $\sqrt{f_{wx}^2+f_{wy}^2}$ to get the torque. But for a pure $f_{xy}$ bivector, the force's plane is parallel to the $w=1$ plane and there is no intersection. The moment calculation is multiplying $f_{xy}$ by $\sqrt{f_{wx}^2+f_{wy}^2}/\sqrt{f_{wx}^2+f_{wy}^2}=0/0$, making it nonsensical. But $f_{xy}$ is still perfectly well defined.
So in 2D mechanics, force is a bivector with three components, $f_{wx}$, $f_{wy}$, and $f_{xy}$, two of them being linear force and one of them being torque. We can generalise this to higher dimensions. In 3D mechanics, force is a bivector with six components: $f_{wx}$, $f_{wy}$, $f_{wz}$, $f_{yz}$, $f_{zx}$, and $f_{xy}$. The first three are the linear force, the last three are the torque.
This same approach can be used to unify linear momentum with angular momentum, linear velocity with angular velocity, and mass with the moment of inertia (where mass turns out to have three components, all of them equal in flat space).
Bivectors in 3D act a lot like vectors - they have three components which you can add componentwise, just like vectors - which is why we can talk about the angular momentum vector and get away with it. It's not quite right. If we apply a reflection, vectors and bivectors act differently, and end up pointing in opposite directions. For this reason, bivectors are sometimes called pseudovectors in the vector-based worldview. However, we can see this really falls apart when we go to a different number of dimensions - in 2D mechanics, the so-called 'angular momentum vector' would have only one component, not two. And in 4D it has six components, not four. It's not a vector; it's part of a bivector.
Adding a dimension like this is using Projective Geometry. Bivectors are found in Geometric Algebra, (also known as Clifford Algebra). The building blocks are conceptually harder, but the resulting physical theory is much simpler. There is one equation of motion that handles both the linear and angular motions together. We don't need to use separate equations and calculations for the linear and angular components, and we halve the number of physical quantities we have to think about at a stroke. And the moment/couple naming problem goes away. It's just an artefact of the coordinate system we use, for purely historical reasons.
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The moment of a single force depends on the point about which moments are taken. So a single force does not have a unique value of moment. That is why the moment is usually written with a subscript denoting the point.
A couple is two equal and opposite forces a set distance apart. The moment produced by a couple Fd is independent of the point about which moments are taken. The moment or torque of a couple is a number unique to that couple.
If these things act upon matter the single force produces an acceleration of the center of mass, and also a rotational effect. A couple cannot affect the motion of the center of mass because the forces add to zero. But they can produce a rotational effect.
Note that even if a body has a fixed axis of rotation its center of mass can still accelerate.
