How to approach dynamics problems?

[cancerman1]

TL;DR Summary: How do I approach the setup of this problem? It seems very different than a setup for a kinematics problem

I'm self studying first year mechanics and am having a hard time with the following problem (screenshot attached). The example is from Intro to Mechanics by K&K.

I'm clear on the steps for setting up and solving kinematics problems (i.e. isolate mass, draw force diagrams, define coordinate system, write equations of motion, etc.), but dynamics problems seem very different. I'd appreciate guidance on how to think about these types of problems

 

[vanhees71]

Study a bit further and learn the Hamilton formalism, and everything gets a lot simpler ;-).

 

[BvU]

Is that really in order at that point in the K&K curriculum ?

 

[Chestermiller]

The issue is that there is more than one way of setting up this dynamics problem, and neither is necessarily preferred over the other. So let's try it your usual way, and see how it can be converted to their way.

The tension in the spring at any time is given by $T=k(r_b-r_a-l)$where $l=r_{b0}-r_{a0}$ where the subscript 0 represent the values at time zero. If we do a force balance on each of the blocks individually, we have $$m\frac{d^2r_a}{dt^2}=k(r_b-r_a-l)$$and$$m\frac{d^2r_b}{dt^2}=-k(r_b-r_a-l)$$These are subject to the initial conditions that $\frac{dr_a}{dt}=v_0$ and $\frac{dr_b}{dt}=0$ at t=0. Ok so far?

 

[cancerman1]

Yes, thank you - your starting point makes it so much clearer versus example 4.7 which starts with the center of mass.

I can logically reason why the setup would be as noted above - a force is applied to block a, and then blocks a and b accelerate from rest but is connected by a spring so that force needs to be accounted for...

 

[Chestermiller]

Yes, thank you - your starting point makes it so much clearer versus example 4.7 which starts with the center of mass.

I can logically reason why the setup would be as noted above - a force is applied to block a, and then blocks a and b accelerate from rest but is connected by a spring so that force needs to be accounted for...

So we're done here, or you'd like me to continue?

 

[cancerman1]

Yes, this should be good for this specific problem.

To generalize a methodology and approach, it's best to start with forces acting on discrete systems and writing up force equations - which is not dissimilar to how one would solve a kinematics problem... would you say that's correct?

 

[PeroK]

Yes, this should be good for this specific problem.

To generalize a methodology and approach, it's best to start with forces acting on discrete systems and writing up force equations - which is not dissimilar to how one would solve a kinematics problem... would you say that's correct?

I would have used the CoM reference frame automatically. I wouldn't have given it a second thought.

That immediately makes the problem symmetric. Conservation of momentum does the rest.

 

[cancerman1]

I would have used the CoM reference frame automatically. I wouldn't have given it a second thought.

That immediately makes the problem symmetric. Conservation of momentum does the rest.

That's helpful framing. I better understand the rationale for the setup in this example. Appreciate it