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I'm new to physics and I've been trying to solve a few high school olympiad questions. I've figured that I approach the questions by analyzing the forces acting on objects and trying to induce something from there, but it gets complicated, and I mean it (whole pages covered in equations). Then I've analytically checked how "they" solve those questions and I've realized their favourite method is to approach questions by considering conservation laws, and it dramatically decreases the amount of time and work required to solve a problem.

I've searched if there is a formal intellectual documentation on this matter but I failed to find one. I'm really looking forward to learn more on this particular situation.

Could you please enlighten me further? (any resource/comment appreciated)

Eg: A and B have masses of 100kg each. For what value of d, will the mass A travel 3 meters then stop?

I don't know, maybe I was unable to write the correct force equations, but when I wrote them and tried to solve the question I miserably failed, due to high amount of equations, whereas the energy approach gets you to the solution in a few lines.

Problem Solution

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  • $\begingroup$ Are we talking about particles moving in 1 spatial dimension? Or are you interested in a more general answer? Which methods of solving differential equations are known to you? $\endgroup$
    – AlmostClueless
    Commented Jun 1, 2021 at 19:43
  • $\begingroup$ @AlmostClueless It could be particles moving in 1 spatial dimension, exaggeratedly hard/long pulley questions, systems of springs, pretty much anything. I've seldom seen them use the so called "Force Approach". I can solve basic ODEs and systems of ODEs. $\endgroup$
    – Efe Zaladin
    Commented Jun 1, 2021 at 20:06
  • $\begingroup$ Could you maybe give an example of what you have seen? $\endgroup$
    – AlmostClueless
    Commented Jun 1, 2021 at 20:17
  • $\begingroup$ @AlmostClueless Added one $\endgroup$
    – Efe Zaladin
    Commented Jun 1, 2021 at 20:40

2 Answers 2

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If you're looking for a very formal mathematical way to codify the evolution of simple, classical physical systems, you can look into an introduction to the Lagrangian formalism and how it compares to the Newtonian formalism.

A particularly interesting case is the case of the double pendulum : A second pendulum attached to the first one, where the mass would be in a single pendulum.

Although the Lagrangian formalism is theoretically equivalent to the Newtonian one, the resulting equations are a lot more manageable:

Here is the example of the simple harmonic oscillator:

https://www.youtube.com/watch?v=KpLno70oYHE

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  • $\begingroup$ Thank you! But I'm aware of the Lagrange mechanics. It would be an overkill for this question wouldn't that be ?(maybe it wouldn't, although I'm aware of it, I haven't actively used it). Also I'm wondering when to decide to use "Energy Approach" etc. (not the advanced topic like lagrangian, hamiltonian, please look at the sample problem) regardless of other better problem solving methods. $\endgroup$
    – Efe Zaladin
    Commented Jun 1, 2021 at 21:00
  • $\begingroup$ Energy conservation principle used as a problem-solving tool: openstax.org/books/university-physics-volume-1/pages/… $\endgroup$
    – Serge Hulne
    Commented Jun 1, 2021 at 21:55
  • $\begingroup$ Thank you again, I can make use of this one $\endgroup$
    – Efe Zaladin
    Commented Jun 1, 2021 at 22:22
  • $\begingroup$ One way to look at it is that forces (and position and speed and acceleration) are vectors (3 dimensional objects, classically), so treating a problem using forces (i.e. using directly Newton's laws instead of the law of conservation of energy) leads (in the case of the motion of as single object) to a set of typically three equations which represent the motion of the object projected on three perpendicular axes. On the other hand, if the applied forces derive from a potential (electric potential, gravity, etc.), then you are dealing with a problem which can boil down to just one dimension. $\endgroup$
    – Serge Hulne
    Commented Jun 2, 2021 at 23:12
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In general, unless you are specifically asked to find equations of motion, a work/energy approach is usually simpler as long as you are given enough information. In this case you have enough information, although the published solution is difficult to follow.

There are two external forces acting on the system of weights and pulleys. One force is gravity which does work $W_1 = mgd$. This work is positive since it accelerates the weights. The other force is friction, which does work $W_2=2\mu mgd$ while the lower weight falls to the ground (because the upper weight moves twice as far as the lower weight) and work $W_3=\mu mg(3-2d)$ thereafter. The work done by friction is negative since it resists the motion of the weights.

So the total work done on the system is

$W_1-W_2-W_3=mg(d-3\mu)$

Since the weights end up stationary, with no kinetic energy, what has happened to the energy that results from this work ? The answer is that it is lost as heat and sound when the lower weight strikes the ground and loses its kinetic energy. Since the speed of the lower weight is always half that of the upper weight, the “lost” energy is one fifth of the total kinetic energy of the system at this point i.e. $\frac 1 5 (W_1-W_2)$. So we have

$mg(d-3\mu)=\frac 1 5 mgd(1-2\mu) \\ \Rightarrow 5d-15\mu=d(1-2\mu) \\ \Rightarrow d(4+2\mu)=15\mu$

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