

Two masses are connected with a string through a
massless and frictionless pulley. The mass m_{1} is 80 Kg
while the mass m_{2} is 125 Kg (see figure). Find the
acceleration of the masses and the tension on the string. Assume all
ideal conditions for pulley and masses. 

The acceleration of the masses is

a)

1.72 m/s^{2
}



b)

9.80 m/s^{2
}


c)

4.90 m/s^{2
}

N

d)

2.15 m/s^{2
}


e)

None of the above.

and the tension on the string is
N 
a) 
956 N 


b) 
1116 N 

c) 
441 N 

d) 
2009 N


e) 
None of the above. 

Solution:
In this problem, since the masses are different, the masses will
accelerate lifting the lighter mass and lowering the heavier mass. Based
on the mass definition of the drawing, the pulley will turn
counterclockwise. For solving this problem, it is necessary to solve for
the acceleration of the two mass and the tension of the string.
It is important to notice that because of the pulley
being ideal, no mass or friction, the tension in the two sides of the
string is the same,
. At the same time, for an ideal string, no stretching
under tension, implies that the magnitude of the acceleration for the
two masses are the same,
.
The direction of the accelerations are opposite to each other because
the mass m_{1} accelerates upward while the mass m_{2}
accelerates downward. 

Therefore, only two equations are necessary to
solve this problem. These two equations will be derived by working the FBDs associated with the individual masses.
FBD Mass m_{1}
The applet at the left shows the development of the free body diagram
to its final form for this mass.
The analysis of the diagram lead to the net force (remember that
forces pointing up are positives and forces pointing down are negative),
The next relation establishing the net force is derived from Newton's
second law,
Equating the two previous relations, it is obtained the first
equation for the problem,
Equation 1
This equation has two unknown; therefore, can not be
solved by itself. A second equation is needed for solving this problem. 

FBD Mass m_{2}
The applet at the left shows the development of the free body diagram
to its final form for this mass.
In this case, the free body diagram provides the following relation
for the net force for the second mass (see the last diagram of the
applet)
Notice that the tension pulls up both masses (in both cases the
string is "holding" the masses). From Newton's second law, the following
relation is obtained,
It is very important to notice that the acceleration
is negative because this mass is accelerating downward. From these two
relations the following equation is derived,
Equation 2 
What about if you can not visualize the situation?
There is not problem, it is only necessary to visualize that the two
masses accelerate in opposite direction. If you mistakenly chose the
wrong direction for the acceleration (clockwise in this case); at the
end, the result for the calculated acceleration is a negative number.
From there, it is concluded that the actual acceleration is in the
opposite direction than the acceleration originally established. 

Now we have two equations and two unknown, such a
system of equations can be solved. Subtracting
Equation 2 from Equation 1 we
obtain for the acceleration,
After solving for the acceleration, this result can
be substituted back into Equation 1 or Equation 2 in order to solve for
the tension of the string. In the applet below, the tension is solved
starting from Equation 1. The final expression for the tension is,
The previous two relations are the algebraic
solutions of the problem.


The numerical solutions are obtained by substituting
the numerical values of the masses and the acceleration of gravity.
Notice that acceleration is positive which means that the direction
of the acceleration is the same as the predicted direction of the
motion, counterclockwise. 
