Systems and number of FBDs....

[fog37]

TL;DR Summary

Situations in which there are more than one system?

Hello,

The system is what we define it to be. The system can include one or multiple objects. These internal objects/bodies can move relative to each other or maintain a fixed distance to form a rigid system. Everything else that is not inside the system is part of the surroundings/environment. The environment can apply a net external force (which may be conservative or nonconservative). Energy or matter (which are kinda the same thing) can transfer in/out of the system's boundary.

Question: if the system is composed of three bodies, we need to draw a free body diagram for each body: one FBD for each internal entity in the system. These internal components may be dependent or independent. If they are dependent, constraint equations will also exist. So far we discussed one system and one environment. Are there situations in which we define more than one system? What would be an example of that? If so, do we just and simply draw a FBD for each system and solve their equations?

Thank you!

 

[Halc]

Not sure what is being asked.

Take the Earth/moon system, which can be modeled as one mass at the barycenter, or two masses, or modeled as many systems of interacting rocks/molecules. OK, that gets needlessly complicated probably, but let's stick to a 2-body system for Earth, but now we add the sun and other planets and their moons as additional systems of 1 to ~60 components each. That's a nice example of multiple systems of multiple components each.

In each system, the internal components interact with each other and their motion constrained accordingly. A FBD of each component would help show a quite accurate picture of the motion of each within the system.

My example is somewhat poor when it comes to interactions with other systems (bodies other than a planet's own moons) since these more distant interactions have only limited effect on the internal interactions, at least for the short term. I can think of other systems with internal interactions which have more major interactions with other systems.

 

[Dale]

if the system is composed of three bodies, we need to draw a free body diagram for each body: one FBD for each internal entity in the system.

This is not usually true. Think of an automobile that has an engine block, pistons, arms, crankshaft, transmission gears, driveshaft, differential gears, axles, wheels, tires, chassis, suspension, body, seats, seatbelts, windows, …. How many times have you seen a physics problem with an automobile where there was one FBD for each internal entity?

There is no fixed rule for the number of FBD’s necessary. The key is to consider what you want to know. Then you need enough FBD’s to produce equations that contain enough information to calculate everything you want to know. The reason that you don’t see hundreds of FBD’s in a typical automobile physics problem is that you don’t care about the related quantities. If you don’t want to know the forces on the camshaft then there is no need for a camshaft FBD.

FBD’s are tools for producing equations, and equations are tools for finding variables of interest. So a big part of the “art” of doing physics is learning to look at a scenario and a set of variables of interest and then figuring out the best way to partition the system to get the easiest equations to solve.

 

[fog37]

This is not usually true. Think of an automobile that has an engine block, pistons, arms, crankshaft, transmission gears, driveshaft, differential gears, axles, wheels, tires, chassis, suspension, body, seats, seatbelts, windows, …. How many times have you seen a physics problem with an automobile where there was one FBD for each internal entity?

There is no fixed rule for the number of FBD’s necessary. The key is to consider what you want to know. Then you need enough FBD’s to produce equations that contain enough information to calculate everything you want to know. The reason that you don’t see hundreds of FBD’s in a typical automobile physics problem is that you don’t care about the related quantities. If you don’t want to know the forces on the camshaft then there is no need for a camshaft FBD.

FBD’s are tools for producing equations, and equations are tools for finding variables of interest. So a big part of the “art” of doing physics is learning to look at a scenario and a set of variables of interest and then figuring out the best way to partition the system to get the easiest equations to solve.

I see. Sometimes we in fact idealize an object with multiple parts as a single point mass system because we neglect all those parts. But in other cases we view the object as composed of multiple internal components.

That said, we seem to always have a single system and a single environment (everything else that is not the system). I have not seen situation in which we set problems up with two entities defined as different systems and the environment...

 

[Dale]

I have not seen situation in which we set problems up with two entities defined as different systems and the environment

Well, you can think of each FBD as defining a system.

 

[jbriggs444]

I see. Sometimes we in fact idealize an object with multiple parts as a single point mass system because we neglect all those parts. But in other cases we view the object as composed of multiple internal components.

That said, we seem to always have a single system and a single environment (everything else that is not the system). I have not seen situation in which we set problems up with two entities defined as different systems and the environment...

Quite often, a large problem can be decomposed into a number of small problems, some simple enough that they can be analyzed in terms of some system which has a limited set of external interactions.

One can examine the differential separately from each piston's interaction with the crank shaft, for instance. Indeed, the major purpose of some components such as the drive shaft is to limit the interactions between one subsystem and another to a simple, well-defined interface.

If one is examining a piece of clockwork, it may make sense to take it one reduction gear at a time.

 

[Mister T]

Are there situations in which we define more than one system? What would be an example of that? If so, do we just and simply draw a FBD for each system and solve their equations?

Of course there are. Just look at any two chapter-end problems. They will likely each describe their own separate system and you can draw a FBD for each system.

 

[hutchphd]

The autmobile is a perfect test case. Do we treat the engine as a subsystem or do we need to worry about each piston individually? The point is to know, in principle, a sufficient granularity or what is necessary to apply knowledge in hand.

 

[jack action]

I think I have the perfect problem for you with what could be considered two systems put together (even though it could also be done as one big system as well): Determining the acceleration of a generic vehicle under an external force.

First, the problem seems simple: $F=ma$ where $m$ is the mass of the vehicle and $F$ is the traction force.

But we know there a rotating parts within the vehicle, beginning with the wheels. These will also add to the total inertia of the vehicle. The "real" solution is to get the mass moment of inertia $I$ of each rotating component $c$ and add all the constraints that link them together, making one big system with multiple FBDs.

If you do that, you will find in the end that the governing equation is now:
$$F = \left(1 + \frac{\sum I_c G_c^2}{mr_w^2}\right)ma$$
Where the subscript $w$ refers to the [reference] wheel, and $G_c = \frac{\omega_c}{\omega_w}$ (better known as the gear ratio for component $c$).

The part in parenthesis is really just another constant that we could set to $\gamma$. From there on, we can set $m_e = \gamma m$ and call it the effective mass. We then get:
$$F=m_e a$$
Which is very similar to our original equation.

And with enough expertise, you can even estimate $\gamma$ without even knowing the details of every component in the system. For example, for a typical automobile driven in a particular gear ratio $G$ (including final drive), we can use this empirical equation:
$$\gamma = 1.04+0.0025G^2$$
$1$ represents the total mass of the vehicle, $0.04$ is related to the typical tires (and everything rotating with them) on a vehicle because $\frac{(km_wr_w^2) (1)^2}{mr_w^2} = k\frac{m_w}{m}$ and the last term takes care of every other component found in a typical vehicle rotating at a different angular velocity.

So there you go: you have two systems that can be modeled independently and then put together in the end.