2 objects connected by a spring - Minimum Force required to move the objects

[EEristavi]

Homework Statement

2 objects with masses m1 and m2 are connected by spring. The kinetic friction coefficient (Between objects and surface) is k. We apply horizontal force - F on object 1 and object 2 starts to move.
What is the Minimum Value of F.

Relevant Equations

Newton's Second Law

On object 2: There are only 2 horizontal forces - Friction and Tension (of the spring).
T = km₂g

On Object 1: There are 3 horizontal forces and the minimum value for F is when:
F - km₁g - km₂g = 0
F = kg(m₁ + m₂)

However, Solution is:
F = kg(m₁ + 0.5 m₂)

Any opinion?

 

[jbriggs444]

However, Solution is:
F = kg(m₁ + 0.5 m₂)

I believe that they are contemplating a solution where the spring between the masses is initially relaxed. A fixed force is applied to $m_1$ and it begins to move under this force at an acceleration that decreases over time as the spring is stretched. Eventually (at a spring force of 0.5 gkm), the acceleration ceases and decelleration ensues. Eventually (at a spring force of 1.0 gkm) $m_1$ comes to a stop with $m_2$ on the verge of slipping.

The provided solution is arguably incorrect, however. First, because it depends on the unstated assumption that the spring is initially relaxed. Second, because better strategies are available if the delivered force is allowed to change signs within the maximum magnitude imposed by F.

Edit: Or to be a complete jerk, one could say that there is no minimum. A force of $-\infty$ will do just fine and is way less than $kg(m_1 + 0.5 m_2)$.

 

[EEristavi]

Edit: Or to be a complete jerk, one could say that there is no minimum. A force of −∞−∞-\infty will do just fine and is way less than kg(m1+0.5m2)

Nice one :D

 

[haruspex]

First, because it depends on the unstated assumption that the spring is initially relaxed. Second, because better strategies are available if the delivered force is allowed to change signs within the maximum magnitude imposed by F.

Third, because we are not told that the static coefficient does not exceed the kinetic.

With regard to varying F, what if F is a constant vector, but necessarily collinear with the separation of the masses? Looks tricky.

Edit: I meant not necessarily.

 

[jbriggs444]

Third, because we are not told that the static coefficient does not exceed the kinetic.

With regard to varying F, what if F is a constant vector, but necessarily collinear with the separation of the masses? Looks tricky.

Ahh yes. A two-dimensional surface. Very nice.