Finding the Minimum Force to Move Two Bars Connected by a Spring

[Saitama]

Homework Statement

Two bars of masses ##m_1## and ##m_2## connected by a non-deformed light spring rest on a horizontal plane. The coefficient of friction between the bars and the surface is equal to ##k##. What minimum constant force has to be applied in the horizontal direction to the bar of mass ##m_1## in order to shift the other bar.

Homework Equations

The Attempt at a Solution

The second bar, ##m_2## starts to move when the force due to spring overcomes the force due to friction on ##m_2## i.e ##F_{spring}=km_2g##. When ##m_2## starts moving, ##m_1## is still moving. Applying Newton's second law on ##m_1##, ##F-km_1g-F_{spring}=m_1a## where F is the minimum constant force asked in the question. I can substitute ##F_{spring}## but that gives me an equation with two variable. How am I supposed to find F from that?

Any help is appreciated. Thanks!

 

[Doc Al]

The second bar, ##m_2## starts to move when the force due to spring overcomes the force due to friction on ##m_2## i.e ##F_{spring}=km_2g##.

Good.

When ##m_2## starts moving, ##m_1## is still moving.

Maybe.

Applying Newton's second law on ##m_1##, ##F-km_1g-F_{spring}=m_1a## where F is the minimum constant force asked in the question. I can substitute ##F_{spring}## but that gives me an equation with two variable. How am I supposed to find F from that?

Force analysis will only get you so far. (The spring force and acceleration of ##m_1## are not constant.)

Hint: Consider the work done by that constant force.

 

[Saitama]

Maybe.

Why maybe?

Hint: Consider the work done by that constant force.

From energy conservation,
[tex]Fx=\frac{1}{2}cx^2+\frac{1}{2}m_1v^2+km_1gx[/tex]
where ##x## is the displacement of ##m_1## and ##c## is the spring constant.

I don't know v but if I put v=0, I do get the right answer but why is v=0?

 

[Doc Al]

I don't know v but if I put v=0, I do get the right answer but why is v=0?

Because you want the minimum force F that will just move ##m_2##.

 

[Saitama]

Because you want the minimum force F that will just move ##m_2##.

Thanks Doc Al!