Net Force and Centripetal Force relation

[fog37]

Hello Everyone,
Question about the centripetal force: the centripetal force is the name given to the component of the net force acting on the object in a direction perpendicular to the object's trajectory and directed towards the center of the osculating circle. This net force component causes the object's trajectory to curve.

In general, in the 2D case, a vector has two components which are referred to a fixed coordinate system (Cartesian x and y axes). When we talk about the net force components, i.e. the component tangent to the object's path and the component perpendicular to the object's path, what frame of reference are we referring to?
Are we considering a local frame of reference centered on the object at that specific point of the trajectory? Is that a fixed coordinate system? Usually we only have a single system of axes with origin O. The position vectors join the origin O and the position of the object at different instants of time.

thanks,
fog 37

 

[FactChecker]

The centripetal force on an object fixed on a rotating disk is always rotating with the disk. The decomposition of the net force into a component perpendicular to the path and tangent to the path does not depend on a choice of coordinate system. The force vector exists and its components exist even before a coordinate system is specified. You should represent them in a coordinate system that makes the calculations easiest.
If you are working on things rotating with the disk, it is probably easiest to use a coordinate system with origin at the center of the disk and rotating with the disk. If you are working on things that are not rotating with the disk, you probably want to determine the coordinates in a coordinate system that is not rotating. You can do that by first representing them in a rotating disk coordinate system and transforming to your chosen non-rotating coordinates.

 

[fog37]

Thanks FactChecker!

I see. Your point "The decomposition of the net force into a component perpendicular to the path and tangent to the path does not depend on a choice of coordinate system. The force vector exists and its components exist even before a coordinate system is specified. " is clear.

I guess I am accustomed to the word "component" in the context of a coordinate system where components are signed numbers referred to the Cartesian axes. But in this case, the components are different...

 

[fog37]

Hi again FactChecker.

One closing comment on solving dynamics problems with multiple objects:
When there are two objects in the situation, we draw a a free body diagram for each object and attach a set of Cartesian coordinates to each FBD. The two sets of Cartesian axes can be oriented differently with origins located in different places (where the two objects are). That implies there are two different coordinate systems at the same time. But in general we fix one single coordinate system and refer all kinematic quantities to that initial coordinate system. How do we resolve this situation, i.e. the fact that we can have multiple coordinate systems at the same time?

Thanks!

 

[jbriggs444]

How do we resolve this situation, i.e. the fact that we can have multiple coordinate systems at the same time?

Pick a coordinate system, express all quantities using that coordinate system, and solve the relevant equations. Then, if desired, transform to a different coordinate system to express the final result.

Free body diagrams need not have coordinate systems. You use coordinate systems when describing vectors in terms of coordinate tuples e.g. (x,y,z), or (r,theta).

It may sometimes be convenient to split the problem up into pieces, using a different coordinate system for each piece (so that some quantity can be ignored or some symmetry exploited). If one does this, it is important to keep track of what frame is being used at any given time and not "frame jump", taking an expression from one frame and calculating with it willy-nilly in another frame without having transformed it first.

 

[fog37]

Thanks jbriggs444. I think this is what I was looking for:

It may sometimes be convenient to split the problem up into pieces, using a different coordinate system for each piece (so that some quantity can be ignored or some symmetry exploited). If one does this, it is important to keep track of what frame is being used at any given time and not "frame jump", taking an expression from one frame and calculating with it willy-nilly in another frame without having transformed it first.

We split the problem up into different parts (see two blocks connected by a rope, one block on an ramp and the other hanging in the air from a pulley) to come up with equations that we then relate to each other via the constraint equations.

 

[FactChecker]

Just to give you an idea of how many coordinate systems there might be in a problem, here are some of the coordinate systems used in simulating the flight of an airplane on a rotating Earth:

• engine force: airplane body coordinates
• aerodynamic forces: wind axis coordinates
• wheel forces: airplane body axis
• front wheel steering force: front landing gear axis system
• inertial reference system: IRS box mounting orientation coordinates
• airplane orientation wrt the level ground below: Locally level coordinates
• airplane position: latitude, longitude, altitude
• effects of rotating Earth: Earth centered, Earth fixed (rotating)

It really becomes more of a bookkeeping task than an engineering task to keep track of it all. And no two things can be combined or added together till you get them into the same coordinate system.

 

[fog37]

Ok. I see. Check the figure below:

In this case each block has its own FBD and the two scalar components equations are written according to the Cartesian axes orientation. Once this is done, we have 4 scalar component equations where some of the terms are related to each other (tension has the same magnitude, net acceleration has the same magnitude, etc.)

The fact is that we solve this problem using multiple coordinate system instead of a single one. Things can be combined regardless of that.

Thanks!