Shifting coordinates for mechanical problems

[aaaa202]

More and more my teacher has talked about how results for mechanical problems are the same no matter what our coordinate system is (though it may be easier to calculate them in some coordinate frames). I must however admit, that I have never really had a clear explanation of what it means to do a problem in different coordinates. Suppose we have a point mass sliding down a frictionless slope and want to find its acceleration. We project the force of gravity onto the direction parallel to the slope and find the acceleration of it. Is this procedure of splitting your force vector into vectors in different directions, what is meant by shifting the coordinate frame? Because if so, how would you ever solve a problem in another frame than the one with axis parallel to the slope and perpendicular to it?

 

[Flipmeister]

Imagine the coordinate system with x parallel to the flat ground and y perpendicular to the flat ground. Could you still find the force along the direction of the slope?

 

[aaaa202]

Yes by projecting gravity perpendicular and parallel to the slope. But isn't that the same as changing coordinate system.

 

[AJ Bentley]

Different co-ordinates can mean a lot of things. Apart from the fact that it's often easier to work with your axes at an angle (Resolving forces parallel and perpendicular to an incline for example), you may find it's easier to use really wild co-ordinates like cylindrical and spherical polar co-ordinates.

For example, in calculating the electric field around a wire we would use cylindrical co-ordinates. In a problem involving satellites and orbits, we'd use spherical ones.

But as to your question about solving a problem in a different frame than tilted along the slope? The answer is to use vector calculations, which can be quite difficult. Physics is often about finding crafty ways to tackle a problem and choosing the right frame is the number one step.