Vertical circular motion

[sdfsfasdfasf]

Homework Statement

x

Relevant Equations

x

So far I believe that speed changes during vertical circular motion, and its very hard to get uniform circular motion that is in the vertical plane.
This is because there is a difference in vertical height between the bottom/top of the circle so at the top the object must have done work against gravity (and hence slowed down). I have found this page http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/cirvert.html which furthers my belief that this is true.

However I keep on seeing different authors and youtube videos where people write down something like "N - mg = Fc at min" and "mg - N = Fc" at max, this doesn't sit right with me because the two v variables are clearly different (unless the object is truly staying at a constant speed, is that even possible in vertical motion?), so the two equations look like rubbish.

Can someone help me out and convince me I have the right idea? If not then I'd love to be told where my mistakes are.
Thank you! My physics course only covers horizontal uniform circular motion but they can always add an extension question where gravity is involved. Thanks.

 

[jbriggs444]

Homework Statement: x
Relevant Equations: x

So far I believe that speed changes during vertical circular motion, and its very hard to get uniform circular motion that is in the vertical plane.

Speed can change during circular motion around a horizontal axis. But it need not do so.

Consider, for example, a bug of negligible mass clinging to the side of a rotating wheel. If the wheel rotates at a uniform rate, the bug will travel in uniform circular motion. If you complain that the wheel is slightly unbalanced due to the presence of the bug, add a second bug positioned opposite the first.

Yes, if you are swinging a sling stone at the end of its strap in a vertical circle then the speed of the stone at the top of the arc will almost certainly be lower than its speed at the bottom. In the absence of non-gravitational tangential force, we can quantify this speed difference by making an energy argument like the one you propose. In the case of a bug on a wheel, there will be tangential force from the tire on the bug to preserve the uniform rotation.

However I keep on seeing different authors and youtube videos where people write down something like "N - mg = Fc at min" and "mg - N = Fc" at max, this doesn't sit right with me because the two v variables are clearly different

Neither of the formulas that you quote here involve $v$. You need to work harder to make a coherent claim.

 

[Orodruin]

Even in the case of non-uniform circular motion, the centripetal acceleration is $v^2/r$. The difference being that there is also a tangential force component and that $v$ is variable because of this.

 

[haruspex]

However I keep on seeing different authors and youtube videos where people write down something like "N - mg = Fc at min" and "mg - N = Fc" at max, this doesn't sit right with me because the two v variables are clearly different (unless the object is truly staying at a constant speed, is that even possible in vertical motion?), so the two equations look like rubbish.

They would be rubbish if the N and Fc are supposed to be constants, or if they are supposed to be scalar variables with consistent sign convention.
And I think you meant and "mg + N = Fc" at max.

$\vec F_{net}= -mg\hat y+ \vec N$, where $\vec F_{net}$ and $\vec N$ are functions of the angle.
Centripetal force is the radially inward component of the net force and $\vec N=-N\hat r$:
$F_c=- \vec F_{net}\cdot \hat r=mg\hat y\cdot \hat r-\vec N\cdot\hat r =mg\hat y\cdot \hat r+N$
where $N$ and $F_c$ are magnitudes.
At the top, $F_c=mg+N$
At the bottom, $F_c=-mg+N$.

 

[Lnewqban]

However I keep on seeing different authors and youtube videos where people write down something like "N - mg = Fc at min" and "mg - N = Fc" at max, this doesn't sit right with me because the two v variables are clearly different (unless the object is truly staying at a constant speed, is that even possible in vertical motion?), so the two equations look like rubbish.

For a constant radius, the value of N in each case depends on the square of the tangential velocity, which evidently shows the greatest value where most of the potential energy has became kinetic energy.

Note that this is applicable to a system using only its initial mechanical energy to complete the vertical loop (a roller coaster car, for example).
It is different for self-propelled bodies that can regulate the tangential velocity (acrobatic airplane, for example).

Please, see:
https://www.physicsclassroom.com/mmedia/circmot/rcd.cfm

https://britishaerobaticacademy.com/how-to-fly-a-loop/

 

[sdfsfasdfasf]

Even in the case of non-uniform circular motion, the centripetal acceleration is $v^2/r$. The difference being that there is also a tangential force component and that $v$ is variable because of this.

So the resultant (centripetal) force varies, correct? As the velocity's magnitude varies throughout the loop.

 

[jbriggs444]

So the resultant (centripetal) force varies, correct? As the velocity's magnitude varies throughout the loop.

Yes. The centripetal component of the resultant force varies with the velocity speed.

 

[Orodruin]

So the resultant (centripetal) force varies, correct? As the velocity's magnitude varies throughout the loop.

Sure, as long as speed varies. As mentioned, if there is self-propellment or similar, it is possible to keep speed constant as well and then the centripetal acceleration does not vary.

 

[sdfsfasdfasf]

Wonderful, thank you so much, Orodruin.