Kleppner - blocks and a compressed spring

• ftang
In summary: The center of mass will maintain a constant speed due to conservation of momentum, while the individual masses will have varying speeds. The total energy (kinetic plus potential) will also be conserved. The solution you found online may have some errors.
ftang
Homework Statement
A system is composed of two blocks of mass m1 and m2 connected
by a massless spring with spring constant k. The blocks slide on a
frictionless plane. The unstretched length of the spring is L. Initially
m2 is held so that the spring is compressed to L/2 and m1 is forced
against a stop. m2 is released at t = 0.
Find the motion of the center of mass of the system as a function
of time.

I understand that when string is back to full Length L, M2 has speed v=0.5L*sqrt(K/m2).
And that is the moment when M1 loses contact with the stop and start moving with M2.

My question 1.
So once M1 leaves the stop, to keep momentum constant, the center of mass should have speed
m2*v/ (m1+m2)
Am I right? I am asking because I read some other solution which states M1 would move at the same speed v as m2, then momentum is not conserved.

My question 2.
Is energy also conserved? In above, if both M1 and M2 moves in speed m2*v/(m1+m2), or both in speed v, in either case energy is not conserved. What is causing energy to be not conservative here??

Thanks a lot!

ftang said:
the center of mass should have speed m2*v/ (m1+m2)
Yes.
ftang said:
M1 would move at the same speed v as m2
Not at the instant m2 loses contact.
ftang said:
if both M1 and M2 moves in speed m2*v/(m1+m2) ... energy is not conserved
No, the mass centre will be moving at that speed. The individual speeds vary, so the total KE will be more.

Thanks Haruspex!
So m1 and m2 moves in different speeds but center of mass moves constant speed as momentum is conserved.
And I believe total energy (kinetic plus string potential) is also conserved. if m1 and m2 moves in individual speeds then all is not lost.

I am surprised at some blunt mistakes in the Kleppner "solution" I found online.

ftang said:
Thanks Haruspex!
So m1 and m2 moves in different speeds but center of mass moves constant speed as momentum is conserved.
And I believe total energy (kinetic plus string potential) is also conserved. if m1 and m2 moves in individual speeds then all is not lost.
Right.

1. What is Kleppner's experiment on blocks and a compressed spring?

Kleppner's experiment involves studying the behavior of two blocks connected by a compressed spring. The blocks are allowed to slide on a frictionless surface, and the spring is compressed by a constant force. The goal is to analyze the motion of the blocks as the spring is released and the blocks start to move.

2. What is the purpose of this experiment?

The purpose of this experiment is to understand the principles of energy conservation and the behavior of systems with multiple objects and forces. It also helps to illustrate concepts such as potential and kinetic energy, and the relationship between force and displacement.

3. What are the key variables in this experiment?

The key variables in this experiment are the masses of the blocks, the spring constant, the initial compression of the spring, and the forces acting on the blocks (such as gravity and the force of the spring). These variables can be adjusted to observe the effects on the motion of the blocks.

4. How can this experiment be applied in real life?

This experiment can be applied in real life to understand the motion of objects connected by springs, such as in a car's suspension system. It can also be used to analyze the behavior of objects in collisions, as well as in engineering and design applications.

5. What are the limitations of this experiment?

One limitation of this experiment is that it assumes an idealized scenario with no external forces or friction. In reality, there will always be some external factors influencing the motion of the blocks. Additionally, the experiment only considers linear motion and does not take into account rotational motion or other complex behaviors.

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