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For example:

(Not a homework question as the answer is below.)

"A bucket full of water is swung in a loop with a radius r. What is the frequency you need to swing it at (revolutions per second) to keep the water from falling out?"

The answer turns out to involve three formulas:

  1. gravitational acceleration = centripetal acceleration

  2. a = ω²r

  3. f = ω/2π

But none of these are really apparent at first sight (to me).

Is there any kind of systematic process or is the answer supposed to just jump out at you?

If you are good at these kinds of problems, what process is your mind going through?

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  • $\begingroup$ My advice is to always start with a formula that includes what you need to find. You need to find frequency, so start with the latter one of your equations. Then you have an unknown in that equation, namely $\omega$, so next step is to find a formula including that. Some experience and "guess work" will help you choose the ideal equations, since some make it easier / faster than others. But without that experience, you might pick a longer way than necessary, but you will still end with the result at some point $\endgroup$ – Steeven Feb 1 '18 at 5:46
  • $\begingroup$ In mechanical problem like that you "just" have to use Newton's law. Then you have to find the forces (gravitational and centripetal here). The $a=\omega^2r$ is more something to know that something to understand and is recurrent in situation involving rotations. Finally $f=\omega/(2\pi)$ is just math and not really phsically important. $\endgroup$ – EigenDavid Feb 1 '18 at 7:07
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    $\begingroup$ However I have to disagree a bit with Steeven's answer. Using equation blindly just because the quantity you are looking for is inside won't help you develop a physical insight. But as John said, nothing is obvious when your begin, and as you go on you'll see that the similar situations arises often and you'll remember what to do. $\endgroup$ – EigenDavid Feb 1 '18 at 7:08
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I really don't think there is any systematic procedure to a problem. As of what I see, you seem to be looking for the formulae to be used rather than questioning the reason behind the happenings. If you start to question why would the water in the bucket fall out and what did the observer in the question do to keep it from falling, those particular equations just pop out of the concepts relating those phenomena.

Once you understand the cause and the effect in a particular situation and you know how to quantify them, the formulae should not be a problem.

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Start with imagining the situation (of swinging that bucket). Realize that the question must be about a rotation in a vertical plane. That the greatest risk of getting wet is when that bucket is above you. Make a sketch, draw relevant forces. Then proceed with calculations. Check if the value is reasonable according to your intuition.

There is an alternative that often works in exam problems, the GUESS-method:

  • list the quantities that are Given
  • identify the Unknown quantity
  • find an Equation that matches these quantities
  • Substitute the values
  • Solve

At school (and also in university), this can deplorably often be done without any physical insight. For example, here it said "with radius $r$". That $r$ works as a cue to use a certain formula. Why do teachers and textbooks include this? One does not even need to mention the word radius, just specify the length of the rope.

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  1. Make a list of what you do know.
  2. If you don't know where to start, start somewhere.
  3. To test a candidate formula, test the boundries by setting the variable to extremes. Does it behave properly at the extremes? Then, you have the right formula.
  4. Use units of measure to guide the structure of the candidate formula (equation). If units of measure are wrong, so is the equation.
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  • $\begingroup$ But what if the length $\ell$ had been specified? And there was the equation with a frequency $f = 2 \pi \sqrt{g/\ell}$? Units are good there. $\endgroup$ – Pieter Feb 1 '18 at 8:33
  • $\begingroup$ If length (l) is meters, and f is 1/sec, then g must be meters/sec/sec, if this is a usable formula, since the cancelled units under the square root must 1/sec/sec $\endgroup$ – Arthur Watkins Feb 2 '18 at 16:40

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