Mass m_{1} is located on mass m_{2} Force F is applied on the upper
mass, analyse the forces acting on the masses and the accelerations.
The maximum friction force acting on m_{1} is
f_{1s} = N_{1}μ_{1s} = m_{1} g μ_{1s}
The maximum friction force acting on m_{2} is
f_{2s} = N_{2}μ_{2s} = (m_{1} + m_{2}) g μ_{2s}
If the force F is gradually increased and f_{2s} > f_{1s}, the friction force f_{1s} is
also increased to the maximum value of f_{1s} = m_{1} g μ_{1s} from this point on,
increase of F will cause mass m_{1} to accelerate and the friction coefficient to decrease to the
kinetic value μ_{1k}. In this case the lower mass m_{2} will not move at all (Because mass m_{2}
should move only by the friction force f_{1} but lower friction force is greater).
From the forces free body diagram in the x direction, mass m_{1} will move when force F overcomes
the static friction force f_{1} .
ΣF_{x} → F − f_{1} = m_{1} a_{1}
F − m_{1} g μ_{1S} = m_{1} a_{1}
(1)
Once force F is bigger then f_{1s} then mass m_{1} will accelerate by:
If we want to enable mass m_{2} to move, then the maximum friction coefficient between mass m_{2}
and the surface is calculated from the forces diagram in the x direction of mass m_{2}, we have:
ΣF_{x} → f_{1} − f_{2} = m_{2} a_{2}
m_{1} g μ_{1s,k} − (m_{1} + m_{2}) g μ_{2s,k} = m_{2} a_{2}
(2)
Note The values of μ is determined from the movement conditions, if the mass is not
sliding then take the static values and if the mass is sliding then take the kinetic values of friction.
a) In order to keep both masses moving together we have to fulfil the condition that a_{1} = a_{2}
and remember that f_{1} is the static force between masses m_{1} and m_{2} and f_{2} is
the kinetic force between mass m_{2} and the surface, then from equations 1) and 2) we get:
