Sign convention for angular and linear acceleration in pulley systems

[rashida564]

I got a confusion about the sings in the angular acceleration. When dealing with system of pulleys, how to define where is the positive and negative direction of the motion and will the choose of positive direction of angular acceleration will effect the positive direction of linear acceleration

 

[BvU]

Do you know about torque as a vector quantity ? Once you are familiar with cross products of vectors you can't go wrong with directions well, almost ...) !

 

[rashida564]

Yeah, I am thinking of doing it slowly by the cross product, but I am not sure how to find the direction of the acceleration

 

[FactChecker]

Summary:: Positive and negative accelaration

I got a confusion about the sings in the angular acceleration. When dealing with system of pulleys, how to define where is the positive and negative direction of the motion and will the choose of positive direction of angular acceleration will effect the positive direction of linear acceleration

Typically, the relationship between the coordinate system axes and the direction of rotation is defined by the right-hand rule. Anything else will cause a lot of confusion.

In your example, you can chose between a couple of coordinate systems that follow the right-hand rule and can have the positive rotations when facing the page to be either clockwise or counterclockwise.

 

[Filip Larsen]

As mentioned already, you want in general to choose positive directions such that the amount of confusion and risk of mistakes are low. Since choice of orientation ends up as signs in equations it is often good to choose such that signs for each term are either all plus or minus (when writing static force equilibrium in the form $\Sigma F_i = 0$).

For simple setups, like the one you showed, where the system readily can be modeled with one degree of freedom (in your case that could be the vertical position of the mass), it would be natural to select positive rotation of each wheel such that it rotates in its positive direction when the mass moves in positive direction (assuming no-slip between string and wheels).

This would of course mean that positive rotation of the two wheels would be opposite each other when compared in their original setup and if this is a problem for some reason, it would also be perfectly OK to select a different orientation as long as you modeled this with proper signs in the equations.

 

[HallsofIvy]

In any problem involving motion or force, the person solving the problem must decide which direction is "positive" and which is "negative". It is customary to make "up" or "right" positive and "down" or "left" negative but that is not necessary. It is customary to make "counter-clockwise" positive and "clockwise" negative for rotational motion but that is not necessary. Of course, once you have made such an assignment, you must be consistent.

 

[FactChecker]

In any problem involving motion or force, the person solving the problem must decide which direction is "positive" and which is "negative". It is customary to make "up" or "right" positive and "down" or "left" negative but that is not necessary. It is customary to make "counter-clockwise" positive and "clockwise" negative for rotational motion but that is not necessary. Of course, once you have made such an assignment, you must be consistent.

One should always use right-hand coordinate systems. So deciding on the positive direction for X and Y in a plane would define the positive direction of Z and of all the rotations.