Understanding Inertial Force in 2nd Order ODEs: Proportional to Acceleration?

[Jhenrique]

When the 2nd order ODE (ay''+by'+cy=0) is taught, is said that the elastic force is proportional to displacement and that the damping force is proportional to velocity, but I never heard the following proposition: "the inertial force is proportional to acceleration (and the constant of proportionality is the mass)". Such proposition/interpretation is correct?

 

[SteamKing]

When the 2nd order ODE (ay''+by'+cy=0) is taught, is said that the elastic force is proportional to displacement and that the damping force is proportional to velocity, but I never heard the following proposition: "the inertial force is proportional to acceleration (and the constant of proportionality is the mass)". Such proposition/interpretation is correct?

Then you must not have heard of Newton's Second Law of Motion:

http://csep10.phys.utk.edu/astr161/lect/history/Newton3laws.html

 

[UltrafastPED]

When the 2nd order ODE (ay''+by'+cy=0) is taught, is said that the elastic force is proportional to displacement and that the damping force is proportional to velocity, but I never heard the following proposition: "the inertial force is proportional to acceleration (and the constant of proportionality is the mass)". Such proposition/interpretation is correct?

It is often used; see http://personal.stevens.edu/~ffisher/me345/sys_dynam_ELECTRICAL_New.pdf

 

[Doc Al]

When the 2nd order ODE (ay''+by'+cy=0) is taught, is said that the elastic force is proportional to displacement and that the damping force is proportional to velocity, but I never heard the following proposition: "the inertial force is proportional to acceleration (and the constant of proportionality is the mass)". Such proposition/interpretation is correct?

When using a non-inertial (accelerating) reference frame, "fictitious" inertial forces are needed in order to apply Newton's laws. See: Fictitious force

 

[D H]

When the 2nd order ODE (ay''+by'+cy=0) is taught, is said that the elastic force is proportional to displacement and that the damping force is proportional to velocity, but I never heard the following proposition: "the inertial force is proportional to acceleration (and the constant of proportionality is the mass)". Such proposition/interpretation is correct?

Did you make that up, or did you read it somewhere? Give us some context, please.

You've received two distinct sets of answers. One is based on Newton's second law, F=ma. The force F in Newton's second law is the net force acting on some body of mass m. However, I've never seen the force F in Newton's second law called the inertial force. It is typically called the net force.

The other set of answers is in the context of a fictitious force. Inertial force is typically taken as another name for fictitious forces in general. The fictitious forces due to rotation all have names:

• The centrifugal force $F_c = -m \vec \omega \times (\vec \omega \times \vec r)$
• The coriolis force $F_C = -2m \vec \omega \times \vec v$
• The euler force $F_e = -m \dot{\vec \omega} \times \vec r$.

What about the fictitious force due to the acceleration of the frame as a whole? That oftentimes isn't given a name, mainly because it doesn't come up so often. When it does come up and people want to give it a name they oftentimes call it *the* inertial force.

This fictitious force does arise often in aerospace engineering and in solar system astronomy. A frame of reference with its origin at the center of planet is an accelerating frame. Aerospace engineers and astronomers don't call the resulting fictitious force the inertial force because the term "inertial force" is generically perceived to be a generic term for any fictitious force. They call this frame acceleration due to other heavenly bodies the "third body effect".

 

[Doc Al]

However, I've never seen the force F in Newton's second law called the inertial force.

Me neither. Thus my assumption that fictitious forces were involved. (In particular, the inertial force -ma, seen with a frame accelerating in a straight line.)

Maybe the OP will provide some context.

 

[D H]

(In particular, the inertial force -ma, seen with a frame accelerating in a straight line.)

Or accelerating on a curved path. The only difference between acceleration on a straight line and accretion along a curved path is that the direction (and possibly magnitude) of the fictitious force due to frame accretion changes over time. At any instant, that fictitious force due to frame acceleration is still just -ma.

Maybe the OP will provide some context.

I second that request.

 

[Doc Al]

Or accelerating on a curved path. The only difference between acceleration on a straight line and accretion along a curved path is that the direction (and possibly magnitude) of the fictitious force due to frame accretion changes over time. At any instant, that fictitious force due to frame acceleration is still just -ma.

Good point!

 

[256bits]

Me neither. Thus my assumption that fictitious forces were involved. (In particular, the inertial force -ma, seen with a frame accelerating in a straight line.)

Maybe the OP will provide some context.

I think you are right about that. The ODE is just the general equation for a mass-damper-spring system, with no external influence, and it is not considered from the view of an accelerating frame of reference. The description in question seems to be a proportionality factor given to the forces from the damper, the spring, and the mass, just to be,possibly, in a manner, consistent.

In the equation, ay''+by'+cy=0, a would represent mass m, b would be damping b, and c would represent spring constant k.
But, by rearranging one obtains by' + cy = -ay", which is strangely similar to what DH wrote about the translational fictitious force equal to -ma. Note that by' and cy are not fictitious but are real forces acting upon the mass. From Newton' s third law equal and opposite forces for 2 bodies interacting, the reaction force from the mass acting upon the spring and damper is equal but in the opposite direction to the force from the spring and damper acting on the mass.

What is your opinion on the above?

 

[UltrafastPED]

Follow the link in #3; here the phrase "inertial" is used for the 2nd order term. In this case the analogy between mechanical and electrical circuits setups is being discussed.

For electrical circuits the 2nd order term is "inductive"; for mechanics it is "inertial". If all of the other terms in the equations have "meaningful names", it is a pedagogical requirement that one be found for the remaining term!

I think the terminology is quite old - it's certainly how I learned in the 1960s.

Note that this has nothing in particular to do with physics - it is a way to describe the terms of the ODE.

 

[Jhenrique]

In the equation, ay''+by'+cy=0, a would represent mass m, b would be damping b, and c would represent spring constant k.
But, by rearranging one obtains by' + cy = -ay", which is strangely similar to what DH wrote about the translational fictitious force equal to -ma. Note that by' and cy are not fictitious but are real forces acting upon the mass. From Newton' s third law equal and opposite forces for 2 bodies interacting, the reaction force from the mass acting upon the spring and damper is equal but in the opposite direction to the force from the spring and damper acting on the mass.

Yeah, this interpretation (the same found here: http://en.wikipedia.org/wiki/Harmonic_oscillators#Damped_harmonic_oscillator) is different of the interpretation for inertial force in my 1st post.

 

[D H]

In the equation, ay''+by'+cy=0, a would represent mass m, b would be damping b, and c would represent spring constant k.
But, by rearranging one obtains by' + cy = -ay", which is strangely similar to what DH wrote about the translational fictitious force equal to -ma. Note that by' and cy are not fictitious but are real forces acting upon the mass. From Newton' s third law equal and opposite forces for 2 bodies interacting, the reaction force from the mass acting upon the spring and damper is equal but in the opposite direction to the force from the spring and damper acting on the mass.

What is your opinion on the above?

My opinion? I can't say on the basis of the Thumperian Principle.

The correct rearrangement is $-k\dot y - cy = a\ddot y$, or $F=a\ddot y$. Your rearrangement, $k\dot y + cy = -a\ddot y$, is mathematically correct but physically meaningless. You have committed a big sin by invoking Newton's third and then adding those equal but opposite forces. One of those equal but opposite forces is acting on the spring, the other on the damper. Adding forces acting on different bodies is invalid.

Yeah, this interpretation (the same found here: http://en.wikipedia.org/wiki/Harmonic_oscillators#Damped_harmonic_oscillator) is different of the interpretation for inertial force in my 1st post.

There is no inertial force in $F=-k\dot x - cx = m\ddot x$, or in the rearrangement to $m\ddot x + k\dot x + cx = 0$.

 

[Jhenrique]

there is no inertial force in $F=-k\dot x - cx = m\ddot x$, or in the rearrangement to $m\ddot x + k\dot x + cx = 0$.

really!?

 

[D H]

Really.

There's a real force from damping (-kv), a real force from the spring (-cx), and a net force (the sum of the two). There is no inertial force here! Net force is not an inertial force. Inertial forces would appear as additional forces on the right hand side: $F_{\text{net}} = \sum -kx - cx +F_{\text{inertial}}$. The only terms on the right hand side are the two real forces.

An inertial force is a fictitious force that results from trying to apply Newton's second law in a non-inertial frame. Strictly speaking, Newton's second law as espoused by Newton is valid only in an inertial frame. However, one can always invent fictitious forces to make it appear that Newton's second law still works in a non-inertial frame. In a sense, inertial forces are an abuse of notation. As is the case with many other abuses of notation, the concept of inertial forces can make for an extremely useful abuse of notation.

As another test of whether inertial forces are present, where's your non-inertial (rotating, accelerating, or both) frame in this problem? You haven't identified one, and presumably you are operating within the context of an inertial frame since the only terms on the right hand side of the net force equation are real forces.

 

[athosanian]

intertial force is considered only in the non-intertial frame, namely, when you consider the motion of a body relative to an accelaration frame. In most cases in the textbook , we analyze the motion of body in the interial frame. the referenced frame is static or moving at a constant velocity. there is no concept of inertial force. Therefore when you are taught by 2nd order eqution, the teacher did not tell you.