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Dynamic Analysis

The mass m1 can slide over a horizontal surface of coefficient of kinetic friction m , the mass m2 is attached to the mass m1 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 m1.

x Direction: (Figure 2 in the applet)

The forces in the x-direction are the tension of the string, T, pulling the mass m1 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 m1 in the x-Direction 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 y-direction are normal force and the weight of the mass m1, . 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 m2, 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

 

 
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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com