M2 - modelling circular motion

Your diagram is fine - although I'd always advise to draw separate free body diagrams of each part, but that's no big deal here.

You know what T is, right. So as before - consider that, for any particular value of $\omega$, there'll be a particular acceleration ( $mr\omega^2$), and that will have to be caused by $T+f$. As you say, it's not known initially what the direction (towards or away from the centre) $f$ is, but it is known that it can only have a magnitude up to $\mu R$.

You seem to be getting frustrated about not really understanding the Physics of it. That's normal and correct - as long as you don't understand it, you need to focus on that. Pay attention to the sentence I bolded above, which is just an extension of the discussion we had in the last thread. If that still doesn't make sense, let me know.

This is just me speculating, but the question mentioned maximum and minimum values of omega. In my mind, two discrete values, one with the frictional force reinforcing the tension in the string, the other with that opposing it, and the rest writes itself.

Your diagram is fine - although I'd always advise to draw separate free body diagrams of each part, but that's no big deal here.

You know what T is, right. So as before - consider that, for any particular value of $\omega$, there'll be a particular acceleration ( $mr\omega^2$), and that will have to be caused by $T+f$. As you say, it's not known initially what the direction (towards or away from the centre) $f$ is, but it is known that it can only have a magnitude up to $\mu R$.

You seem to be getting frustrated about not really understanding the Physics of it. That's normal and correct - as long as you don't understand it, you need to focus on that. Pay attention to the sentence I bolded above, which is just an extension of the discussion we had in the last thread. If that still doesn't make sense, let me know.

I am attaching the diagrams and workings. They come up with the correct answers.

In diagram 1 why is friction acting away from the centre and if the angular speed was slower than the minimum what would actually happen to the particle and the forces? Also, what would actually happen if the speed exceeded the maximum calculated? Can you do it diagrammatically for me and talk me through it or is it something that requires higher maths/physics?

The part of the chapter I'm on is followed by 2 more bits: "Problems involving non-horizontal forces" and "Circular motion with non-uniform speed". Do you think that my lack of understanding will be clarified at all with this work or do I need to do some other reading on the current topic before moving on?

If the angular speed was slower than the minimum, then the resultant force on the particle (the one on top) is $> mr\omega^2$, right. That means it has an acceleration towards the centre that's bigger than $r\omega^2$. That means it will actually fall towards the hole. Friction will be trying to prevent that (relative motion), but it will fail as it can't get any bigger.

Something that's often helpful is to consider extreme values, in Mechanics, because our physical intuition can confirm for us whether our equations are right. Example: let's imagine that $\omega$ is ridiculously small and then ridiculously large. If it's small what happens? Clearly the particle rapidly slides towards the hole if there isn't a significant rotation. If it's large, i.e. fast, then it will move outwards, which could also be seen as moving to a larger 'orbit'.

I just remembered actually doing this experiment in real life. We took a fine wire, attached small masses to each end, threaded the wire through a long glass tube, and then you can hold it up and whirl it around Here's a simple diagram: http://www.batesville.k12.in.us/physics/phynet/mechanics/circular%20motion/labs/cf_and_speed.htm
So like in this question, one mass sits at the bottom and the other one rotates in a (hopefully) horizontal circle. There is no friction here (apart from air) so it's not exactly the same, but it's something. You see that the faster you force it to rotate, the larger the circles the rotating mass makes.


"Problems involving non-horizontal forces" is a title suggesting that they are not dealing with non-horizontal circular motion; it will probably just be a matter of resolving forces in the direction towards the centre of the circle (radial). Nothing particular there (will be on inclined planes and so on).
"non-uniform speed" - this can be a bit more demanding. What does *not* change is the formula for centripetal acceleration, and that you have to find out how the existing forces cause that acceleration. What does change is that, with changing speed, there is also acceleration in the transverse direction. There is no change in the Physics here, it's just more complicated (for example, if you use energy conservation you have to notice that the kinetic energy of the particle rotating is changing).

It's difficult to answer the 'do I need to do more reading'. I can see from your attempts at answering that you know how to construct the diagrams and equations. The more difficult non-constant-speed case is just the same principles applied to a more difficult situation. More reading/watching vids about circular motion would be a good idea to help you to see how it works in different ways and get you to feel more comfortable with it (but - is the problem about how circular motion works, or about how friction works?)

Thankyou. You have been really helpful.

Since my last post I have been doing a bit of reading from a different book called "Understanding Mechanics". It's sort of getting me there. In the original book it didn't actually explain how the particle was kept on a circular path but the other one did and now it seems that now that bit of fog has gone I can take thing on board better. Fingers crossed.

I am an adult learner teaching myself hence so many questions to TSR. Despite it being headbangingly difficult for me I just love mechanics. Weird. I notice from your profile that you are a maths teacher. After circular motion I have only one more chapter for AQA M2. You might advise me not to, but I'd really like to try M3 later. The Amazon reviews tell me that the AQA book M3 is out of date for the current syllabus. Can you recommend a suitable textbook?

I have come across a website that says that the following topics are AQA M3:
Relative motion - relative velocity and displacement, Collision in 1 dimension - impulse and Newton's Law, Impulse and momentum, Further Trajectories, Projectiles on Inclined Planes and Using Dimensions.
I currently have Applied Mathematics 1 (Bostock & Chandler), Understanding Mechanics (Sadler & Thorning).

Thanks again.

Unfortunately my experience has been almost entirely with CIE syllabus, so I'm not the person to ask about the different modules in AQA or the best textbooks for them.

As to books, I used to use Sadler and Thorning now and then for the exercises, although never as a class text. As I recall it's not bad. Bostock and Chandler is very good, have used it extensively, just be aware that like the other B&Cs it's sometimes covering quite a bit of stuff not needed in the current A-level syllabi.

Self-teaching means you need to plan things out yourself; the way I'd do it is start from the syllabus, then use the exam papers to see exactly what the syllabus means in terms of types of questions. Even if you can't answer them immediately, you can find out what the syllabus is actually talking about. Then you can map it into which parts of your textbooks you need to study (for example, there may be whole sets of exercises in a certain chapter in Bostock and Chandler which, while interesting, would never come up on your exam).

I do realise it's easier to just buy the textbook which covers exactly what you need and go through it chapter by chapter. If that exists (someone else maybe knows), then great, it's easier, but if not you'll have to be careful and use the syllabus/exam papers as your guide.

I certainly wouldn't 'advise you not to' do M3. Sure the higher numbered modules are harder, and the top end of Further Maths Mechanics is at least slightly hard for everyone not in the 'gifted' category, but there is no reason to be scared, just give it a go. M3 looks fine.