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.

 

[Nickie]

Yes, the procedure of splitting the force vector into components along different directions is one example of shifting the coordinate frame. Essentially, shifting the coordinate frame means changing the perspective from which we view a problem. In the example you provided, the force of gravity can be represented as a vector pointing straight down, but we can also break it down into components along the slope and perpendicular to it. This allows us to simplify the problem and make it easier to solve.

To solve a problem in a different coordinate frame, we would need to transform our original problem into the new coordinate system. This can be done using mathematical equations and principles such as vector addition and rotation. For example, if we wanted to solve the same problem in a coordinate system where the slope is at an angle, we would need to rotate our original coordinate system to match the new angle.

Shifting the coordinate frame can also be useful in situations where the original coordinate system is not the most convenient or intuitive. For instance, in some cases, using polar coordinates instead of Cartesian coordinates can make the problem easier to solve. It is important to note that the results of the problem will still be the same, regardless of the coordinate system used. It is simply a matter of finding the most efficient and effective way to solve the problem.

In summary, shifting the coordinate frame in mechanical problems refers to changing the perspective from which we view and solve the problem. This can be done by breaking down vectors into components along different directions or by transforming the original coordinate system into a new one. Ultimately, the goal is to make the problem more manageable and to find the most efficient way to solve it.