Force=mass times acceleration, or time derivative of momentum?

In summary, F=ma is still the fundamental law for a material system, but F=dp/dt is a more general law that takes into account momentum flux.
  • #1
Loren Booda
3,125
4
Which is more generally corrrect,

F=ma

or

F=dp/dt

?
 
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  • #2
Aren't they mathematically equivalent?
 
  • #3
Some particles, such as photons, don't have mass but they still have a momentum. F=ma won't work in these cases.
 
  • #4
Depends on the situation, are you dealing with an impulse, where you need to consider momentum transfer, or a longer duration event, where acceleration acts over a longer time duration. F=ma or F=dp/dt is situational.
 
  • #5
Aren't they mathematically equivalent?

Not in general, if you consider the product rule.
 
  • #6
F= ma is the same as F= dp/dt only if m is a constant. F= dp/dt is more general.
(In fact, one can consider "rate of change of momentum" to be the DEFINITION of force.)
 
  • #7
Loren Booda said:
Which is more generally corrrect,

F=ma

or

F=dp/dt

?
Let = mv. Then

[tex]F = \frac{dp}{dt}[/tex]

is a definition of F. F = ma is an equality between the quantities F, m and a when m is constant.

Incorrectly assuming that F = ma is a definition has gotten people really mixed up when going to relativity.

Pete
 
  • #8
My question arises from the rocket problem, where the object expends mass.
 
  • #9
Hi, Loren:
The crucial difference between the rocket+remaining fuel system and "normal" systems is that it is an "open" system (or geometric control volume) that loses momentum-carrying particles over time, whereas material systems (or material control volumes) keeps all momentum-carrying particles through all times.

To take a silly example:
Have a frictionless table of finite length, and let your system be that part of a book which happens to remain on top of the table, the book sliding along with some constant velocity V.
As the book gets to the edge, the momentum of your chosen system decreases because the parts of the book that travels beyond the table edge is not within your chosen system.
That is, material particles leave your observed system and carry their own momentum with themselves. This is called momentum flux.
But the correspomding decrease of momentum in your system cannot be related to any net force acting upon the material particles remaining in your system!

Thus, for open systems, we need to take care of the momentum flux term in order not to get wrong answers.

It is for this reason that the rocket equation looks somewhat differently than the version of Newton's 2.law for a MATERIAL system.
 
  • #10
pmb_phy said:
Let = mv. Then

[tex]F = \frac{dp}{dt}[/tex]

is a definition of F. F = ma is an equality between the quantities F, m and a when m is constant.

Incorrectly assuming that F = ma is a definition has gotten people really mixed up when going to relativity.

Pete

Just a comment to this:
Pre-relativistically speaking, your fundamental laws for material systems were those of F=ma and mass conservation, and F=dp/dt was a derived law.
It is Einstein who deserves the credit for bringing about a more productive conceptual switch.
 
Last edited:

1. What is the definition of force?

Force is a physical quantity that describes the interaction between two objects. It is measured in Newtons (N) and is equal to the mass of an object multiplied by its acceleration.

2. How is force related to mass and acceleration?

The relationship between force, mass, and acceleration is described by the equation Force = mass x acceleration. This means that the greater the mass of an object, the more force is needed to accelerate it.

3. What is the significance of the time derivative of momentum?

The time derivative of momentum is a measure of how an object's momentum changes over time. It is significant because it helps us understand the forces acting on an object and how they affect its motion.

4. How is force calculated in real-world applications?

In real-world applications, force is typically calculated by measuring the mass and acceleration of an object and using the equation Force = mass x acceleration. Other factors such as friction and air resistance may also need to be taken into account.

5. How does force affect the motion of an object?

Force is responsible for changing the motion of an object. If a force acts on an object, it will accelerate in the direction of the force. The greater the force, the greater the acceleration will be. In the absence of external forces, an object will continue moving at a constant velocity.

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