Max Force Applied to System of Two Masses Connected by Rope w/o Breaking

[vu10758]

Two masses accelerate along a flat horizontal surface. They are connected by a rope. There is no friction between M_2 and the surface but friction between M1 is described by mu_k. the rope connecting M1 and M2 (M2 is right of M1) breats at a tension T_o. What is the maximum force F that can be applied to the system (pulling M2) that can be applied to the system without the rope breaking.

This is what I did so far.

I drew the free body diagrams for both. For M1, I have Normal force pointing up, graviyt down, friction to the left, and tension to the right.

For M2, I have gravity down, normal force up, F to the right, and T to the left.

For M1

x) T-f_k = (m1)*a
y) N - M1*g = 0

For M2

x) F-T = M2*a
y) N2- M2*g = 0

So I got T = m1a + f_k
and F = T + M2 * a

I plugged things in and got

F = (M1 + M2)a + mu_k * N1

However, I know this is the wrong answer. I am suppose to express F in terms of T_o somehow. What am I suppose to do from here?

 

[Fermat]

You will have two expressions for the acceleration of each mass (F=ma).
Divide one equation by other and eliminate the acceleration a. Solve for T_o.

 

[vu10758]

If I divide I get

(T-f_k)/(F-T) = m1/m2

What do I do now? I know that I cannot use a in the final answer but both of the relationships I got from the free body diagrams involves a.

 

[Fermat]

Sorry, I should have said solve for F, in my last post.

(T-f_k)/(F-T) = m1/m2
(m2/m1)(T-f_k) = F-T
F = T + (m2/m1)(T-mu_k.m1)

 

[vu10758]

Thanks very much.