Newton's second law clarification - derivative, constant mass

In summary, Newton's second law states F = d(mv)/dt as this law is valid only for constant mass systems it is also written as F = md(v)/dt. But let's suppose the mass was not a constant, then the derivative of the law would become F = m'v + mv' (where by ' I mean derivative, m' = dm/dt), would that be right? If the law is valid only for constant mass why are we writing it inside the derivative and only after that we take it out as a constant?
  • #1
nitai108
14
0
Newton's second law states F = d(mv)/dt as this law is valid only for constant mass systems it is also written as F = md(v)/dt.
But let's suppose the mass was not a constant, then the derivative of the law would become F = m'v + mv' (where by ' I mean derivative, m' = dm/dt), would that be right?
If the law is valid only for constant mass why are we writing it inside the derivative and only after that we take it out as a constant?
 
Science news on Phys.org
  • #2
I doubt your statement that Newton's second law holds good only for constant mass system. I have successfully used it in past for variable mass systems like sand falling on/from a moving vehicle, and he rocket propulsion problems. I think that is what troubled you.
 
  • #3
nitai108 said:
Newton's second law states F = d(mv)/dt as this law is valid only for constant mass systems it is also written as F = md(v)/dt.
But let's suppose the mass was not a constant, then the derivative of the law would become F = m'v + mv' (where by ' I mean derivative, m' = dm/dt), would that be right?
If the law is valid only for constant mass why are we writing it inside the derivative and only after that we take it out as a constant?

Newtons' second law IS valid for variable mass systems. Whatever (or whoever) gave you the idea that it is not, was just wrong.

You are right that for a variable mass F = m'v + mv'.

There are many practical examples of the second law applied to variable mass systems. For example, a rocket burning fuel, or a conveyor belt being loaded by material falling onto it.
 
  • #4
Thanks for the replies!
I was reading Feynman Lectures and I searched Newton second law to see if it could be used in a variable mass system, of course wikipedia came first and there are a few notes which state that the law is valid only for constant mass systems (note 20 21 22 http://en.wikipedia.org/wiki/Newton's_laws_of_motion#cite_note-plastino-19 )
Could you please elaborate why did they write this?
 
  • #5
It depends if you call the introduced momentum flux as a force, or if you regard it as..just flux of momentum.

If you regard momentum flux as a force, then Newton's 2 law holds good in variable mass problems.

However, in classical mechanics proper, we generally regard momentum flux as just that, bevcause it is, really simpler to model lots of problems from this perspective.

I wrote this thread some years ago, distinguishing between the "material system" and the "geometrical system".

I advise you to read through it:
https://www.physicsforums.com/showthread.php?t=72176
 
  • #6
It's only valid for closed (constant mass) systems, otherwise you need to introduce an external force to account for the loss of momentum. Imagine a moving particle which loses mass with time - using Newton's 2nd law it will accelerate to conserve momentum. Now analyse the same problem in the frame where the particle is initially at rest - you will find it stays at rest. This breaks Galilean relativity. The rocket equation is a simple modification which generalises the law to variable mass systems.
 
  • #7
F = d(mv)/dt

When m = constant, F = m dv/dt because the derivative of a constant * variable = constant * derivative of the variable

When m is not constant, F = m dv/dt + v dm/dt by the Chain Rule of derivatives.

This second equation is fundamentally rocket science.
 
  • #8
SteamKing said:
F = d(mv)/dt

When m = constant, F = m dv/dt because the derivative of a constant * variable = constant * derivative of the variable

When m is not constant, F = m dv/dt + v dm/dt by the Chain Rule of derivatives.

This second equation is fundamentally rocket science.
And is the "v" you differentiate here the same as the "v" multiplied with dm/dt?
And, is your F=0?
And if not, how do you determine it?
 
  • #9

FAQ: Newton's second law clarification - derivative, constant mass

1. What is Newton's second law?

Newton's second law is a fundamental principle of classical mechanics that describes the relationship between an object's mass, acceleration, and the applied force. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

2. What is the derivative in Newton's second law?

The derivative in Newton's second law refers to the mathematical concept of the derivative, which is used to express the relationship between an object's acceleration and the applied force. It represents the rate of change of acceleration with respect to time.

3. How is constant mass related to Newton's second law?

According to Newton's second law, the mass of an object remains constant regardless of the applied force. This means that the acceleration of an object is only affected by the magnitude and direction of the force, not its mass.

4. Can Newton's second law be applied to objects with varying mass?

Yes, Newton's second law can be applied to objects with varying mass as long as the mass is changing at a constant rate. In this case, the derivative of mass with respect to time would also need to be included in the equation.

5. How does Newton's second law relate to real-life situations?

Newton's second law is applicable to a wide range of real-life situations, such as the motion of cars, airplanes, and projectiles. It helps to understand the relationship between the applied force and the resulting acceleration of an object, which is essential in fields such as engineering and physics.

Similar threads

Back
Top