

The mass m_{1} can slide over a
horizontal surface of coefficient of kinetic friction m
, the mass m_{2} is attached to
the mass m_{1} by an ideal string
and mass less pulley. 

In order to solve the dynamics of this problem, the
acceleration of the masses and tension of the string must be calculated.
For a non stretching string, the acceleration of both masses are the
same. The method for solving for these quantities requires the
identification of the FBD for both masses.
Let us start with the mass m_{1}.
x – Direction: (Figure 2
in the applet)
The forces in the xdirection are the
tension of the string, T, pulling
the mass m_{1} to the right; and, the
kinetic frictional force resisting the
motion to the right by pulling the block to the left. Thus, the net
force in the horizontal direction is
. Accordingly with
Newton's second law, this net force is related to the acceleration of
the mass m_{1} in the xDirection by
Putting together the two relations for the net force
in the horizontal direction,
Equation 1 
y – Direction: (Figure 3
in the applet)
The forces in the ydirection are
normal force and the weight of the mass
m_{1},
. Therefore,
the net force in the vertical direction is
. Newton's
second law applied to this direction results in
The previous relations for the net force in the
vertical direction implies the following equation,
Equation 2
Considering Equation 1
and Equation 2, there are three unknown,
the tension, the friction, and the acceleration. However, the kinetic
frictional force is related to the normal force by the additional
equation,
Equation 3
Substituting the results
Equation 3 in Equation 1, the final
equation for the FBD (Figure 4 in the
applet) for the mass on top of the table is obtained,
Equation 4 

Continuing with the mass m_{2}, for
which there are vectors only in the vertical direction (Figure
1 in the applet) . Thus, the net force in the vertical direction
is the vector addition between the tension, pulling the mass up, and the
weight of the mass, pulling the object down,
Equation 5
Newton's second law applied to this mass vertical
component is,
Substituting this result in
Equation 5,
Equation 6
The two unknown of the problem can be obtained by
solving the system of equations, Equation 4
and Equation 6. First let us solve
for the acceleration, 


The previous applet shows the algebra
involved in the calculation of the acceleration. The result is
Equation 7
The next applet shows the calculation of the tension
on the string final expression
The final expression for the Tension is
Equation 8


