How can one relate jerk and force?

[Dark85]

TL;DR

I have a doubt regarding a paragraph on Wikipedia on the relation between force and jerk.

I was exploring how to find expressions for a changing force, hence I began with the simplest case where a body moves with a constant jerk and has a constant mass. Here is what I did:


The relation between force and momentum is given by:
$$
\vec{F} = \frac{d(m\vec{v})}{dt}
$$

So, as I understand, one can derive an expression for force in terms of jerk or higher derivatives. For constant jerk and constant mass, I derived an expression for jerk to be(if my understanding is correct):

$$
\vec{F}(t) = m\vec{a}(t)
$$

$$
\frac{d\vec{F}(t)}{dt} = m\frac{d\vec{a}(t)}{dt}
$$

Let constant jerk $ = j$

$$
d\vec{F}(t) = m\vec{j}dt
$$

integrating both sides

$$
\vec{F}(t) = m\vec{j}t
$$

However I found this statement on the Wikipedia article for [jerk](https://en.wikipedia.org/wiki/Jerk_(physics)#Force,_acceleration,_and_jerk) and it says that:

Which I did not quite understand. How are no forces associated with higher derivatives of acceleration? Does the expression for force in terms of momentum not allow this (i.e. not apply for higher derivatives) as such as this? If so, why?

 

[Baluncore]

How are no forces associated with higher derivatives of acceleration?

F = m a ;
The rate at which a changes, determines the rate at which F changes.
That holds for all derivatives of a.

 

[Dark85]

F = m a ;
The rate at which a changes, determines the rate at which F changes.
That holds for all derivatives of a.

So the Wikipedia article is wrong? I understand that it is not completely trustable source but I could not find a discussion on the relation between jerk and force anywhere else as far as I looked.

 

[Ibix]

To restate what @Baluncore has said, $F=ma$ and it has no dependence like $F=ka\frac{da}{dt}$ nor $F=ma+k\frac{da}{dt}$. That's also what the Wiki article is saying.

All you have done is written $a=\int j\,dt$ and substitute it into $F=ma$. That isn't introducing an explicit dependence on jerk, it's just calculating $a$ on the fly.

 

[jrmichler]

The Wikipedia article is correct, BUT you need to be very aware of the difference between simplified physics calculations and real world calculations. Simplified physics calculations assume rigid point masses, while the real world has elastic masses with physical size.

If you apply a force to a rigid point mass, the mass accelerates. This is true regardless of the jerk, which can be infinite. Infinite jerk merely means that the force is applied suddenly, without ramping up.

If you suddenly apply a force to a real world system with physical size and elasticity, then deformations and oscillations occur. This is exactly what the Wikipedia article is stating. An extreme example is a Slinky toy.

Engineers working with high performance servomotor systems need to be very aware of jerk. A suddenly applied force (or torque) causes oscillations. Those oscillations can be large enough to make the system unusable. The solution is to apply the force with a finite jerk, where the jerk duration is one cycle of the oscillation frequency.

 

[A.T.]

How are no forces associated with higher derivatives of acceleration?

It just means that to compute $F$, you don't need to know the derivatives of $a$, if you already know $a$ itself.

If you don't know $a$ itself, and have to express $a$ as function of some other known quantities, that doesn't mean that $F$ has a dependency on those quantities in general. It's just a case specific relationship, while $F=ma$ is a general relationship.