What happens to Newton's 2nd law if there is a changing mass?

[mitcho]

I understand that force is equal to the time derivative of momentum, or d(mv)/dt. Then what happens if the velocity is constant and only the mass is changing. Does this mean there will be a force. If so, in what direction since I am assuming it is still a vector. Also, what if the mass and velocity are changing, does this make some kind of "2nd order" force?
Any help would be appreciated.
Thanks.

 

[Pengwuino]

So what you have now is $$\frac{d}{dt}(m(t)v(t))$$. This is simply a product rule, nothing beyond that. You'll simply have a more complicated problem.

For example, if you had a simple harmonic oscillator, your 2nd law would become:

$$\frac{d}{{dt}}(m(t)\dot x(t)) = - kx(t)$$

which upon doing the differentiation simply gives

$$\dot m(t)\dot x(t) + m(t)\ddot x(t) = - kx(t)$$

Of course, you'll need information on the functional relationship of m with respect to t to solve this. I haven't given this much thought but that does not look like an easy problem

 

[Curl]

I don't understand how you can have a change in mass. Newton didn't anticipate mass annihilation, so I'm not sure what becomes of the law.

 

[bp_psy]

I don't understand how you can have a change in mass. Newton didn't anticipate mass annihilation, so I'm not sure what becomes of the law.

You can have a system in which you just ignore a part of the mass.For example you ignore the fuel that was ejected in a rocket if you are interested only in the motion the rocket. F=dp/dt so both mass and velocity can vary.This does not mean that mass is annihilated only that you can sometimes ignore some of it.

 

[PJC01]

If the mass is changing, then the force must also be changing in order to maintain a constant acceleration. This is because Newton's second law states that force is equal to mass multiplied by acceleration (F=ma). So if the mass is changing, the acceleration must also change to maintain a constant force.

In the case of a constant velocity and changing mass, there will not be a force in the direction of motion, since there is no change in velocity. However, there may be a force in a different direction if there are other external forces acting on the object.

If both the mass and velocity are changing, then there will be a "second order" force, as you mentioned. This is because the acceleration is changing, which means the force must also change to maintain a constant acceleration. This second order force would be in addition to any other external forces acting on the object.

Overall, the key point to remember is that Newton's second law applies to the net force acting on an object, which means that any changes in mass or velocity will result in a corresponding change in force to maintain a constant acceleration.