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In our dynamics class, it is highly recommend to us to use D'Alembert's approach rather than the traditional approach to solving dynamic problems. For example, consider the problem of a block resting on a surface (we know the cof): Rather than calculating the frictional force of the block on the slope and comparing that to the component of weight acting down the slope, we draw the free body diagram including the inertial force and directly solve for a, which will tell is if the block is moving or not. My question is why this method is recommended over the traditional approach? It seems the answer is usually with regards to making it easier to solve more complex problems, but I don't quite understand why this trivial manipulation of Newton's second law makes it much easier to solve problems?

I'm sorry if this has been asked before, but I couldn't quite find a satisfactory answer online!

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  • $\begingroup$ What do you mean by the 'traditional approach'? Your example of 'calculating the frictional force of the block on the slope and comparing that to the component of weight acting down the slope' is a static approach, not dynamic since it does not consider inertia. $\endgroup$
    – atom44
    Jul 20 '16 at 10:26
  • $\begingroup$ Quote from en.wikipedia.org/wiki/D%27Alembert%27s_principle, "The advantage is that, in the equivalent static system one can take moments about any point (not just the center of mass). This often leads to simpler calculations because any force (in turn) can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation (sum of moments = zero)" $\endgroup$ Jul 20 '16 at 13:18
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I don't really understand how D'Alembert's principle "helps" in solving the problem in your OP, but I assume your teacher wants you to practise using it in simple situations before you try more complicated ones.

The real value of working in a non-inertial coordinate system is when such a system makes it easier to describe the problem.

For example, suppose you want to model the dynamics of an aircraft. Planes can move in any direction in 3-dimensional space, and the nose of the plane can also be pointing in any direction - not necessarily in the same direction that the plane is moving (see any video of an aerobatics display for examples!)

The easiest way to described the forces created by the plane's engines and control surfaces (rudder, tailplane, ailerons) is to use a coordinate system fixed to the plane. Since this coordinate system may be accelerating and/or rotating relative to the ground, it is not an inertial coordinate system and you have to include the d'Alembert forces depending on how it is moving.

Another example is the dynamics of rotating machinery. Suppose you want to model the vibrations of the blades of a wind turbine, while it is rotating at constant angular velocity. The easiest way to set up the model is in a coordinate system that rotates with the blades, including the d'Alembert forces. In general the blade vibration frequencies will not have any simple relation to the speed of rotation of the whole windmill, and they will depend on the rotation speed. In fact, because of the Coriolis forces on the system, even in the rotating system the blades don't vibrate back and forth in simple harmonic motion in a straight line, like the systems you study in a first dynamics course. Each point on the blade actually moves in around an elliptical path whose center is at the "non-vibrating" position. Trying to describe that motion, and set up the equations of motion, in an inertial coordinate system fixed relative to the ground would be much more complicated than using a coordinate system relative to the blade itself. And if you use a computer model to solve the dynamics of the system, the easiest way to visualise the results is to make an animation in the same rotating coordinate system - i.e. to display what you would see if you were "sitting on the blade" as it rotates.

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