What is the (higher order) time derivative of centripetal acceleration?

In summary, the conversation discusses the time derivative of centripetal acceleration and its potential extension to higher order time derivatives. The concept of finite infinite-time derivative is also mentioned and the validity of the calculation is confirmed. However, it is noted that dimensional analysis alone does not guarantee the correctness of the result and a basic differentiation is necessary to confirm it.
  • #1
TheCanadian
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Just using basic dimensional analysis, it appears the time derivative of centripetal acceleration is ## \vec{r} \omega^3 ##, but this intuitive guess would also extend to higher order time derivatives, no? Implying:

## \frac {d^n \vec{r}}{dt^n} = \vec{r} \omega^n ##

It seems to follow from the general result shown in Thorton/Marion pg 390 (attached) when considering rotating bodies in a fixed frame. I assume ## \vec{Q} ## is any vector, even ones that are the result of a higher order time derivative of an initial vector. The concept of finite infinite-time derivative just seems like an odd concept to me when considering real objects, but I guess the geometry of the situation allows it. But to confirm, is anything posted here incorrect?
 

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  • #2
TheCanadian said:
Just using basic dimensional analysis, it appears the time derivative of centripetal acceleration is r⃗ ω3r→ω3 \vec{r} \omega^3 , but this intuitive guess would also extend to higher order time derivatives, no? Implying:
Dimensional analysis allows that as a possibility, but it does not make it true. If we start with the scalar form ##r\omega^2## we see the time derivative is ##r\dot\omega^2+2r\omega\dot{\omega}##.
 
  • #3
haruspex said:
Dimensional analysis allows that as a possibility, but it does not make it true. If we start with the scalar form ##r\omega^2## we see the time derivative is ##r\dot\omega^2+2r\omega\dot{\omega}##.

Thank you for pointing out that I forgot to do a basic differentiation. It is much appreciated.
 

Question 1: What is the definition of centripetal acceleration?

Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It is always directed towards the center of the circle and its magnitude is given by the equation a = v^2/r, where v is the velocity of the object and r is the radius of the circular path.

Question 2: What is the first order time derivative of centripetal acceleration?

The first order time derivative of centripetal acceleration is the rate of change of the centripetal acceleration with respect to time. This is also known as the jerk or the rate of change of acceleration. It is given by the equation j = dv/dt = a/t, where v is the velocity, a is the acceleration, and t is time.

Question 3: What is the second order time derivative of centripetal acceleration?

The second order time derivative of centripetal acceleration is the rate of change of the first order derivative, or the rate of change of jerk. This is also known as the snap or the rate of change of jerk. It is given by the equation s = da/dt = j/t = a/t^2, where a is the acceleration, t is time, and j is the jerk.

Question 4: What is the third order time derivative of centripetal acceleration?

The third order time derivative of centripetal acceleration is the rate of change of the second order derivative, or the rate of change of snap. This is also known as the crackle or the rate of change of snap. It is given by the equation c = d^3a/dt^3 = s/t = j/t^2 = a/t^3, where a is the acceleration, t is time, s is the snap, and j is the jerk.

Question 5: How are higher order time derivatives of centripetal acceleration used in physics?

Higher order time derivatives of centripetal acceleration are used to describe the motion of objects in circular paths more accurately. They can also be used to analyze the stability and oscillations of systems in rotational motion. In addition, these derivatives are used in equations of motion for rotational systems, such as the Euler-Lagrange equations and the Newton-Euler equations.

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