TBR stresses Rotational analogs to linear motion

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plzNOCarribbean

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I saw the AAMC PS topics list and although Uniform Circular Motion and Centripetal Acceleration was mentioned, there was nothing on the rotational analogs of linear motion (angular velocity and angular acceleration). Do we need to know these?

the reason I ask is because chapter II of the TBR physics book on newtonian mechanics goes pretty in depth on it. I want to make sure I understand, particularly because this was the one section I struggled with in undergrad physics 1 and i felt overwhelmed reading the chapter today. Any advice would be appreciated.

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Ok, so I saw this on a previous thread by BONOZ but it didn't adequately answer my ? so I am re-posting on the topic.

In TBR (2.8a) a child riding on a merry-go-round jumps off the horse and starts walking towards the center of the ride. As he moves, the acceleration he feels:

B) increases, because his radius from the center decreases
D) decreases, because his radius from the center decreases

I initially though B because Ac=v^2/r but there is also the equation with ac=(w^2)(r).

w=V/r can re-arrange wr=V; based on these equations, I see that by decreasing the radius, the tangential velocity would also decrease. by going back to the equation Ac=V^2/r if the velocity is decreasing, then Ac must be decreasing, and if the radius is decreasing, the Ac must be increasing. These two contradict one another, but the V term is squared so could this be a way that we safely know that that answer is correct?

TBR stated that we use A=w^2r because the angular velocity is constant anywhere on a spinning circle. OK?.... but based on the equation w=V/r, wouldn't the angular velocity INCREASE as r decreased, therefore complicated the situation once more since now if we go back to the Ac=w^2 (r), now the angular velocity term is increasing while r is decreasing...ughh

sorry, I know that was long and winded but I hope someone can follow my logic and clarify this for me.

lastly, basically, for all of these rotational analogs to linear motion, is it safe to say that they are ALWAYS constant or does it depend on what the question is asking? I mean, is this always the case? is it something that everyone knows to assume? I just can seem to understand this because by modifying the r, why wouldn't we be affecting the ANGULAR VELOCITY AND ANGULAR ACCELERATION....
 
w=V/r can re-arrange wr=V; based on these equations, I see that by decreasing the radius, the tangential velocity would also decrease. by going back to the equation Ac=V^2/r if the velocity is decreasing, then Ac must be decreasing, and if the radius is decreasing, the Ac must be increasing. These two contradict one another, but the V term is squared so could this be a way that we safely know that that answer is correct?

TBR stated that we use A=w^2r because the angular velocity is constant anywhere on a spinning circle. OK?.... but based on the equation w=V/r, wouldn't the angular velocity INCREASE as r decreased, therefore complicated the situation once more since now if we go back to the Ac=w^2 (r), now the angular velocity term is increasing while r is decreasing...ughh

You have to understand what is being kept constant when you use this sort of logic. How do you know that by decreasing the radius, the tangential velocity would decrease UNLESS angular velocity was constant? There is no contradiction because what you're describing is two different situations, one in which radius is constant, and one in which tangential velocity is constant.

Whenever you have a mathematical relationship like a = bc, a is dependent on BOTH b and c. And if b and c are both variables, simply increasing b doesn't necessarily increase a. For example, if I tell you that I have a rectangle whose length is 5, and another rectangle whose length is 10, can you tell me which rectangle has greater area? On the other hand, you can tell me that a circle with a radius of 10 has 4 times the area of a circle with a radius of 5. Pi is constant. In this specific problem, angular velocity and angular acceleration are constant.

lastly, basically, for all of these rotational analogs to linear motion, is it safe to say that they are ALWAYS constant or does it depend on what the question is asking? I mean, is this always the case? is it something that everyone knows to assume? I just can seem to understand this because by modifying the r, why wouldn't we be affecting the ANGULAR VELOCITY AND ANGULAR ACCELERATION....

No, each problem is different. Modifying r could modify angular velocity, but that depends on the tangential velocity. What you know for sure, though, is that changing r must change at least one of those two (angular and tangential velocity). Angular velocity is just dθ/dt, just like linear velocity is just dx/dt (Of course, it doesn't have to be "x" axis..., but I think you know what I mean).
 
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