A large box of mass M is pulled across a horizontal, frictionless surface by a horizontal rope with tension T. A small box of mass m sits on top of the large box. The coefficients of static and kinetic friction between the two boxes are μs and μk, respectively.

Question

Find an expression for the maximum tension Tmax for which the small box rides on top of the large box without slipping.

Express your answer in terms of the variables M, m, μs, and appropriate constants.

Attempt:

T = ma + mg

Ff = μ Fn
   = μ (m)(a)

a = Ff/(μ (m))

T = (Ff/μ) + mg

I am completely stuck. I am not sure how to approach this.

Anonymous · Accepted Answer

T = gμs(M+m)

Scott · Answer

the surface is frictionless, so T is providing acceleration to the boxes

the accelerating force is transferred to the small box by friction between the boxes

at some point (as T increases), the frictional force will not be enough to provide the necessary acceleration and the small box will slip

that's the point you're solving for

a = T / (M + m)

m g μs = m T / (M + m)
... solve this for T