Atwood's Machine consists of two objects (of mass m1 and m2) hung over a pulley. Treating the pulley as massless and frictionless and treating the rope as massless and of fixed length, find an expression for the size of the acceleration of the objects.
Solution: Atwood's Machine is an excellent example of a problem that can be dealt with using many different definitions of the system. We will solve it using three different methods to showcase the advantages and disadvantages of various choices.
Method 1
Systems: Object 1 as a point particle and object 2 as a separate point particle system. Each is subject to external influences from the earth (gravity) and the rope (tension).
Model: Point Particle Dynamics.
Approach: We begin by treating each object separately. We construct free body diagrams:
where the tension force on each of the objects is exactly the same size.
It is important to recognize that the tension force acting on each object will be the same only because of four separate assumptions stated in the problem:
- The pulley is massless (we treat the massive pulley case in [Atwood's Machine Revisited].
- The pulley is frictionless.
- The rope is massless.
- The rope does not stretch.
If any of these assumptions is unreasonable, the results we obtain would be wrong.
Using the free body diagrams as a guide, we write Newton's 2nd Law for each object. We consider only the y direction, since the x direction simply yields the equation 0 = 0. Because everything is in the y direction, we work with the magnitudes of the forces and explicitly include signs (rather than including lots of tedious "y" subscripts).
\begin
[ T - m_
g = m_
a_
]
[ T-m_
g = m_
a_
]\end
To finish this problem, we must use the constraint on the acceleration. The accelerations of objects 1 and 2 are constrained by the fixed length of the rope to be equal in size and opposite in magnitude. If the masses are known, it makes sense to substitute in such a way as to find the positive acceleration. In our case, we arbitrarily substitute for a2 using:
\begin
[ a_
= -a_
]\end
giving:
\begin
[ T - m_
g = -m_
a_
]\end
Eliminating the tension from the system of equations then gives:
\begin
[ a_
= \frac{(m_
-m_
)g}{m_
+m_{1}}]\end
Since the accelerations are the same size, we can characterize the acceleration of the system as:
\begin
[ a = |\frac{(m_
-m_
)g}{m_
+m_{1}}|]\end
Method 2
A second method is to use a "math shortcut" by defining an unphysical system that replicates the mathematical features of the problem.
System: Mathematical construction including the two objects and the rope connecting them. The system is subject to external forces from gravity on each block.
Model: Point Particle Dynamics.
Approach: This method uses the fact that when a massless, frictionless pulley is involved, the pulley has essentially no role in the system except to change the direction of forces. Thus, the dynamics of Atwood's Machine are completely equivalent to the dynamics of the system sketched in [physical representation] here:
Of course, this system is anything but physical. The important point is that the mathematics are identical. This trick of "straightening" the system is often useful when a massless, frictionless pulley is involved. Note that we have chosen to draw the system in the x direction instead of the y direction to remind us that gravity will not pull in the same direction on each mass! Thus, the free body diagram for the system looks like:
and the resulting form of Newton's 2nd Law is:
\begin
[ m_
g - m_
g = (m_
+m_
) a ]\end