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__average speed__remains constant as it continues to circulate.

Any thoughts and/or confirmation?

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- Thread starter John Mohr
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In summary, the conversation discusses the concept of circular motion in the horizontal and vertical directions. It is mentioned that in the vertical case, there may be a net force that either assists or resists the motion, causing the object to slightly speed up or slow down. The conversation also addresses the terminology and calculations used for analyzing vertical circular motion, with some disagreement among participants. Finally, there is a discussion about a diagram and its accuracy in representing the forces at different points in the circular path.

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Any thoughts and/or confirmation?

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How would you analyze it for the more accurate case you have identified?

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Now, it is true that in a vertical circular motion, like the one of a mass attached to a string, if the only forces are the weight and the tension then the speed will not be constant.

That does not mean you can't have a uniform vertical circular motion, it would just require some compensation forces in order to to have a constant speed/centripetal force.

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1. A mass going around a circle at constant speed has constant angular momentum.

2. If the circle is vertical, there is the varying external torque of gravity acting on it.

3. Therefore to keep the angular momentum and hence the speed constant, another external torque must compensate for the gravity torque. This torque can only come from the string.

4. However, a string attached to a fixed point can exert only a radial force which means zero torque. So ##\dots## how does one get out of this?

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John Mohr said:But I'd often would see in books, notes, and the internet that we assume constant velocity for the vertical case.

That is not consistent with my experience. In most of the treatments and examples I see, the motion in vertical circles is not uniform circular motion.

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Thank you to everyone for your comments. I’m sorry I didn’t include a diagram earlier. I got home and finally had a chance to draw a diagram of what I mean.Chestermiller said:How would you analyze it for the more accurate case you have identified?

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Can you give an example? It depends on the scenario. For a spinning wheel the speed on the circumference can be constant in the vertical case.John Mohr said:But I'd often would see in books, notes, and the internet that we assume constant velocity for the vertical case.

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Let's see your Newton's 2nd Law equations for this.John Mohr said:Thank you to everyone for your comments. I’m sorry I didn’t include a diagram earlier. I got home and finally had a chance to draw a diagram of what I mean. View attachment 278771

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Your diagram is faulty. You show two external forces, tension T (blue) and weight W (green). OK. Then you show net force FJohn Mohr said:Thank you to everyone for your comments. I’m sorry I didn’t include a diagram earlier. I got home and finally had a chance to draw a diagram of what I mean. View attachment 278771

At the due East position there should be a read arrow pointing down and to the left. There is none. Why not? Similarly at the due West position.

At the due North position again you show no red arrow. Why not? Both the tension and the weight point down, therefore their red arrow sum must also point down. Similarly at the due South position except that the blue and green arrows point in opposite directions. However, they cannot cancel each other out because there must be net centripetal force to make the mass follow the circular path.

Having said all that, I am not sure under what conditions you drew your diagram. One possibility is that the mass is given a good kick at the due South position and more than enough speed to go around the loop. The other possibility is that the string is replaced by a light rod with some kind of driving mechanism at the center that causes the mass to go around the circle at constant speed. That's the Ferris wheel analogue. The third possibility of a mass at the end of a string going around the circle of fixed center at constant speed is not a possibility for reasons that I explained in post #4.

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$$m a \ddot{\theta}=-m g \sin \theta \; \Rightarrow \; \ddot{\theta}=-\frac{g}{a} \sin \theta.$$

The solution is an elliptic function and the angular velocity, ##\dot{\theta}## is for sure not constant.

You can see this of course already in the small-angle approximation, i.e., for ##|\theta|\ll 1##, where you can approximate ##\sin \theta \simeq \theta##, from which you get

$$\ddot{\theta}=-\omega^2 \theta, \quad \omega=\frac{g}{a}.$$

The solution is

$$\theta(t)=A \cos(\omega t + \varphi),$$

and the angular velocity is

$$\dot{\theta}(t)=-A \omega \sin(\omega t+\varphi).$$

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But, perhaps to start, I thought a visual example might be helpful to explain what I've seen before. I know there is a whole lot more going on with the physics on this example but I'm hoping the general idea will be seen. When we see gymnasts doing 'giants' on a high bar...

...it can be seen that they slow down at the top of their swing. In a thought experiment too - if one was to pump your legs hard enough on a playground swing and you could avoid having the chain becoming slack and falling down, one could swing so hard that you'd go right around. But, just beyond the minimum speed you would likely be slower at the top then speed up at the bottom. Of course, in this scenario you would need a different mechanism where the chains attach so they wouldn't wrap around.

For this diagram, I tried to think about the 2D vector addition and how long the Fnet would be. Though my scales are approximate I know it's not perfect. But the Fnet is supposed to be the sum of T and W is either is translated to the tip of the other.kuruman said:Your diagram is faulty. You show two external forces, tension T (blue) and weight W (green). OK. Then you show net force F_{Net}in red. Not OK. The net force must be the vector sum of the blue and green arrows, reasonably drawn to scale. It is not.

I know also that I didn't draw in the Fnet for all positions of the rotating object (N, S, E, and W) But certainly these could be shown too. The only two which I'd think wouldkuruman said:At the due North position again you show no red arrow. Why not?

I think vanhees71 may be onto something here, but the math is something I'd need to brush up on or learn. Apologies, I teach at the high school level and it's been awhile. :)vanhees71 said:I think with "vertical circular motion" you mean a pendulum.

On this document practice problem #1talks about a ball swung in a vertical circle at 5 m/s. And on this page if one searches for the word 'constant' it appears four times with the context being vertical circular motion and objects going at a constant speed.A.T. said:Can you give an example?

Sorry Chestermiller, I wasn't sure what you were wanting for this. Maybe I could analyze one or two of the positions on the path to show this. I'll try to put this together.Chestermiller said:Let's see your Newton's 2nd Law equations for this.

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It is "swung", so it doesn't just move unter gravity.John Mohr said:On this document practice problem #1talks about a ball swung in a vertical circle at 5 m/s.

From the linked page:John Mohr said:And on this page if one searches for the word 'constant' it appears four times with the context being vertical circular motion and objects going at a constant speed.

It is "whirled", so it doesn't just move unter gravity. It also states that the speed changes.

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Hey that's right A.T., your right! I didn't see that before. Very cool - so it does say something about this. And from my own reference too.A.T. said:View attachment 278860

...It also states that the speed changes.

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Are you implying that at high schools nowadays you don't treat the mathematical pendulum anymore? It's sooooo sad!John Mohr said:I think vanhees71 may be onto something here, but the math is something I'd need to brush up on or learn. Apologies, I teach at the high school level and it's been awhile. :)

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You don't need to analyze one or two positions. I'm asking you to write down the force balances in the radial and tangential directions (or, equally acceptable, the vector force balance in terms of unit vectors in the radial and tangential directions). Or, equally acceptable, the conservation of energy equation involving the kinetic energy and potential energy of the object. I would like to see some equations and a token effort to model this problem.John Mohr said:Sorry Chestermiller, I wasn't sure what you were wanting for this. Maybe I could analyze one or two of the positions on the path to show this. I'll try to put this together.

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Hello Chestermiller, I've given the problem some thought and here's what I've come up with.Chestermiller said:You don't need to analyze one or two positions. I'm asking you to write down the force balances in the radial and tangential directions (or, equally acceptable, the vector force balance in terms of unit vectors in the radial and tangential directions). Or, equally acceptable, the conservation of energy equation involving the kinetic energy and potential energy of the object. I would like to see some equations and a token effort to model this problem.

How does this seem to you?

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John Mohr said:Hello Chestermiller, I've given the problem some thought and here's what I've come up with.

View attachment 278904

How does this seem to you?

What goes up must slow down!

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Very good. So we have$$mr\left(\frac{d\theta}{dt}\right)^2=T+mg\sin{\theta}$$andJohn Mohr said:Hello Chestermiller, I've given the problem some thought and here's what I've come up with.

View attachment 278904

How does this seem to you?

$$mr\frac{d^2\theta}{dt^2}=-mg\cos{\theta}$$

Do you agree?

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So you're not interested in seeing what the solution to these equations looks like?John Mohr said:

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Sorry Chestermiller, I didn't mean to stop our conversation prematurely. Please, yes, I would be interested in the solution you have proposed.Chestermiller said:So you're not interested in seeing what the solution to these equations looks like?

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