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why is a constant force and constant linear speed requirements for uniform circular motion? I think I understand the idea of constant force:

enter image description here

this is what comes to mind when I think of force changing but would like to confirm that that's the reason. However I do not understand why linear speed/angular speed have to be constant too. I would just imagine the the particle would move faster but still in the same uniform circle. so why is constant speed a requirement for uniform circular motion?

additionally is constant force a requirement for centripetal (not necessary for it to be uniform) motion? and if yes then why so because if the image i provided is correct, then the particle still moves in a circle of decreasing radius

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  • $\begingroup$ Do you understand why, in uniform circular motion, the acceleration has a constant magnitude? $\endgroup$
    – DanDan面
    Commented Mar 30 at 9:00
  • $\begingroup$ if the centripetal force is constant, then yes acceleration would have a constant magnitude. $\endgroup$ Commented Mar 30 at 9:09
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    $\begingroup$ That reasoning is a bit circular (no pun intended. well, maybe a little bit of a pun intended.) If we only start with the assumption of uniform circular motion with speed $v$, can you characterize the acceleration associated with this motion? This is a purely kinematics question $\endgroup$
    – DanDan面
    Commented Mar 30 at 9:20
  • $\begingroup$ thats my question why does it have to have a constant acceleration magnitude/force $\endgroup$ Commented Mar 30 at 11:21

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First of all, "uniform circular motion" means by definition that $v$ is constant. In order to have a constant speed $v$ along (!) the circular trajectory, we need that the acceleration $a$ is perpendicular to the speed (as otherwise $v$ would change in magnitude instead of just the direction). The last thing we have to see is that $a$ is constant, but as you already pointed out, this must be the case for circular motion because if not, we would have some net acceleration inwards (if $a$ becomes larger) so that the circle is not "closed".

Note as an aside, that if the acceleration would be constant but not pointed perpendicular to $v$, we could decompose the acceleration in a part that is perpendicular (constant) and one that is parallel to $v$. We would then have a motion that is a combination of uniform circular motion and accelerated lineair motion along the trajectory.

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  • $\begingroup$ so if there is a parrallel acceleration, then the ball does not move in a "perfect circle" ? $\endgroup$ Commented Mar 30 at 11:18
  • $\begingroup$ and if so, how can a ball speed up in a constant radius circle? $\endgroup$ Commented Mar 30 at 11:24
  • $\begingroup$ Indeed, the ball would not move in a perfect circle. In order for a ball to speed up in a constant radius circle (think of a person rotating a ball hanging from a rope, something you can do yourself), the force (in this case perpendicular to $v$ ) would need to increase. As an additional practical example. Consider a person sitting in the car while it is making a turn. If the car turns too fast, the person will slide to the outside (possibly against the door). This sliding outward causes the extra force (frictional force). $\endgroup$
    – OonyXx
    Commented Mar 30 at 11:38
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Suppose you are moving and a constant force is being applied to you. You can resolve the force into two components. One is parallel to your motion, and the other perpendicular to it.

Suppose you have just the component parallel to your motion. Then you travel in a straight line. If the force is in the direction you are moving, it speeds you up. If it is against your direction, it slows you down. If it happens to be $0$, it doesn't change your speed.

Suppose you have just the component perpendicular to your motion. It deflects you from a straight line into a curved trajectory. It changes the direction you travel. If the force is always perpendicular to your direction, the force changes direction along with you.

If the perpendicular force is constant and the parallel force is $0$, two things happen. Your speed never changes. And you are deflected into a uniform curve. That is, every section of the curve looks like every other section. It never gets sharper or flatter. If you think about it, you will see the only curves like that are straight lines and circles.

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