Spring momentum conservation problem

[member 731016]

Homework Statement

I have successfully solved the problem below by assuming that momentum is conserved and that there is an inelastic collision occurring between the masses. However, I am wondering whether the momentum being conserved is a valid assumption that I have made.

Relevant Equations

$E_i = E_f$
$\vec p_i = \vec p_f$

For this problem,

The reason why I am not sure whether it is a valid assumption whether momentum is conserved because during the collision if we consider the two masses to be the system, then there will be a uniform gravitational field acting on both masses, and a spring force that is acting upwards. Therefore, there will be two external forces acting on the system. The only reason I can think of for momentum being conserved in this case is if the forces acting on the both the masses acted over such a short time interval that there was no change in the momentum due to the forces.

However, if we define the system as everything, the two masses, the spring, and the source of the g-field, then I believe everything is internal force pairs so momentum is conserved.

If someone please knows whether momentum is conserved is a valid assumption and why, that would be greatly appreciated!

Many thanks!

 

[kuruman]

The only reason I can think of for momentum being conserved in this case is if the forces acting on the both the masses acted over such a short time interval that there was no change in the momentum due to the forces.

This reasoning is correct. Over the time interval that the masses stick together they are not displaced appreciably in the gravitational field.

 

[topsquark]

Homework Statement: I have successfully solved the problem below by assuming that momentum is conserved and that there is an inelastic collision occurring between the masses. However, I am wondering whether the momentum being conserved is a valid assumption that I have made.
Relevant Equations: $E_i = E_f$
$\vec p_i = \vec p_f$

For this problem,
View attachment 330735
The reason why I am not sure whether it is a valid assumption whether momentum is conserved because during the collision if we consider the two masses to be the system, then there will be a uniform gravitational field acting on both masses, and a spring force that is acting upwards. Therefore, there will be two external forces acting on the system. The only reason I can think of for momentum being conserved in this case is if the forces acting on the both the masses acted over such a short time interval that there was no change in the momentum due to the forces.

However, if we define the system as everything, the two masses, the spring, and the source of the g-field, then I believe everything is internal force pairs so momentum is conserved.

If someone please knows whether momentum is conserved is a valid assumption and why, that would be greatly appreciated!

Many thanks!

The momentum of the 1.0 kg mass is clearly not conserved. Neither is the momentum of the 2.0 kg mass. But if you take the 1.0 kg mass, the 2.0 kg mass, and the spring (which is ideal, and thus massless) to be your system, then the momentum is conserved, (If you want to be picky, throw the Earth into this and use the Newtonian gravitational potential energy.)

-Dan

 

[haruspex]

The momentum of the 1.0 kg mass is clearly not conserved. Neither is the momentum of the 2.0 kg mass. But if you take the 1.0 kg mass, the 2.0 kg mass, and the spring (which is ideal, and thus massless) to be your system, then the momentum is conserved, (If you want to be picky, throw the Earth into this and use the Newtonian gravitational potential energy.)

-Dan

That doesn’t really help. The issue is the time interval over which momentum is to be considered.
If you take the time up until the dropped mass reaches its lowest point, clearly momentum of the two mass+spring system is not conserved. Including the Earth is not being picky, it's essential.
To avoid that, we can use a very short time interval, making the assumption that the coalescence is achieved quickly.
In between these extremes, we could model the coalescence as a spring of high constant during compression and zero constant in relaxation.

 

[member 731016]

This reasoning is correct. Over the time interval that the masses stick together they are not displaced appreciably in the gravitational field.

The momentum of the 1.0 kg mass is clearly not conserved. Neither is the momentum of the 2.0 kg mass. But if you take the 1.0 kg mass, the 2.0 kg mass, and the spring (which is ideal, and thus massless) to be your system, then the momentum is conserved, (If you want to be picky, throw the Earth into this and use the Newtonian gravitational potential energy.)

-Dan

That doesn’t really help. The issue is the time interval over which momentum is to be considered.
If you take the time up until the dropped mass reaches its lowest point, clearly momentum of the two mass+spring system is not conserved. Including the Earth is not being picky, it's essential.
To avoid that, we can use a very short time interval, making the assumption that the coalescence is achieved quickly.
In between these extremes, we could model the coalescence as a spring of high constant during compression and zero constant in relaxation.

Thank you for your replies @kuruman, @topsquark and @haruspex!