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Sundown444
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I have been wondering, simple question, really: What is the relationship between momentum and centripetal acceleration, if there is one? Is there a relationship in terms of velocity, maybe, or is there none whatsoever?
Not sure if i get what you are asking here, but consider a point mass in motion along some curve: every single point P on the curve can be locally approximated by a circle of radius [itex]R_P[/itex] (see osculating circle), so at every point you can compute a value for your centripetal acceleration [itex]a_c = \frac{v^2}{R_P}[/itex], so if you remember that linear momentum is [itex]q = mv[/itex] you'll note that if velocity increase by a factor of 2, linear momentum will increase by the same factor, while centripetal acceleration will increase by a factor of 2^2=4.Sundown444 said:I see. Well, I have one more question. Centripetal Acceleration is proportional to the square of the velocity divided by the radius. How is the square of the velocity related to the velocity in momentum, if it is?
By a power of 2.Sundown444 said:How is the square of the velocity related to the velocity in momentum
mastrofoffi said:Dr.D's answer is neat, but if you're looking for something more I think you may want to clarify what you mean by being related; both of them have a dependency on velocity, yes, but that would make every cinematic quantity somehow related with each other, thus making your question a bit vague.
Not sure if i get what you are asking here, but consider a point mass in motion along some curve: every single point P on the curve can be locally approximated by a circle of radius [itex]R_P[/itex] (see osculating circle), so at every point you can compute a value for your centripetal acceleration [itex]a_c = \frac{v^2}{R_P}[/itex], so if you remember that linear momentum is [itex]q = mv[/itex] you'll note that if velocity increase by a factor of 2, linear momentum will increase by the same factor, while centripetal acceleration will increase by a factor of 2^2=4.
Likewise, if you change the centripetal acceleration acting on the body by a factor k (without changing its istantaneous radius of curvature), its speed, and therefore its linear momentum, will change by a factor [itex]\sqrt{k}[/itex]
No, how can the velocity be independent from itself? I mean, it is the same velocity we are talking about, it just plays a different role in the 2 quantites, momentum being linear in v while centripetal acceleration is quadratic. In fact, as I showed you, with fixed mass and trajectory a change in centripetal acc implies a change in linear momentum and viceversa.Sundown444 said:I see. Well, beyond the square root of velocity in centripetal acceleration and such, there wasn't much else I was asking for, if anything at all. So, the velocity squared in centripetal acceleration is independent from the velocity in momentum, I take it?
mastrofoffi said:No, how can the velocity be independent from itself? I mean, it is the same velocity we are talking about, it just plays a different role in the 2 quantites, momentum being linear in v while centripetal acceleration is quadratic. In fact, as I showed you, with fixed mass and trajectory a change in centripetal acc implies a change in linear momentum and viceversa.
If what you are asking is "is the [itex]v^2[/itex] in the centripetal acceleration directly derivable from the linear momentum?" then for sure you can find some fancy way to derive it like that(consider a body under action of centripetal force, note that F = dq/dt, etc..) but it would be hardly meaningful with respect to the standard way of defining those quantities.
The relationship between momentum and centripetal acceleration is that they are both related to circular motion. Centripetal acceleration is the acceleration directed towards the center of the circle, while momentum is the product of an object's mass and velocity. In circular motion, centripetal acceleration is necessary to keep an object moving in a circular path, and momentum is conserved as the object moves.
Centripetal acceleration does not directly affect an object's momentum. However, it is necessary for an object to maintain circular motion, which in turn affects the object's momentum. As an object moves in a circular path, its velocity is constantly changing, and this change in velocity results in a change in momentum. Therefore, centripetal acceleration plays a role in the change in an object's momentum.
Yes, an object can have momentum without experiencing centripetal acceleration. Momentum is a property of an object that depends on its mass and velocity, while centripetal acceleration is a type of acceleration that occurs in circular motion. An object can have momentum in any type of motion, not just circular motion.
No, centripetal acceleration does not affect the direction of an object's momentum. The direction of an object's momentum is determined by its velocity, which is tangential to its circular path. Centripetal acceleration acts towards the center of the circle and is always perpendicular to the object's velocity, so it does not affect the direction of the object's momentum.
The equation for centripetal acceleration, a = v^2/r, is related to the equation for momentum, p = mv, through the velocity term. In both equations, velocity is a crucial factor. In the equation for momentum, velocity is multiplied by mass, while in the equation for centripetal acceleration, velocity is squared and divided by the radius of the circular path. This shows that both momentum and centripetal acceleration are dependent on an object's velocity in circular motion.