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Atwood's Machine consists of two objects (of mass m1 and m2) hung over a pulley.

    Part A

    Treating the pulley as massless and frictionless and treating the rope as massless and of fixed length, find an expression for the accelerations of the two masses.

    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 two methods to showcase some of the pitfalls of constructing a system when a pulley is involved.

      Method 1

      Systems:

      Object 1 as a point particle and object 2 as a separate point particle system.

      Interactions:

      Each system is subject to external influences from the earth (gravity) and the rope (tension).

      Model:

      Point Particle Dynamics.

      Approach:

      Interactions:

      External forces from gravity on each block.

      Model:

      Point Particle Dynamics.

      Approach:

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      Part B

      Find the tension in the rope in terms of the object masses and the acceleration of gravity.

      Solution

      Systems:

      As in Part A, Method 1.

      Interactions:

      As in Part A, Method 1.

      Model:

      As in Part A, Method 1.

      Approach:

      Part C

      Part A makes it clear that the system of the two objects has an accelerating center of mass (provided the masses are not exactly equal). Using the results of Method 1, we can find the acceleration to be:

      Show that this center of mass acceleration is consistent with the sum of the external forces on the system.

      Solution

      Systems:

      The two objects, the pulley and the rope.

      Interactions:

      The system is subject to external forces from gravity and from the axle of the pulley. We will also have to consider the pulley as an independent point particle system acted upon by the rope (tension) and by the axle.

      Method:

      Point Particle Dynamics.

      Approach:

      Diagrammatic Representation

      A truly physical approach that incorporates both objects into one system must also include the pulley. We begin with a picture and free body diagram for that system:

      It is key to realize that although the pulley is massless and frictionless, it is acting to support the system through a force exerted on it by the axle. A simple way to see this is to consider what would happen to the system if the axle was removed.

      Mathematical Representation

      We can now write Newton's 2nd Law for the system:

      where acm is the acceleration of the system's center of mass.

      To find FA, we must use the free body diagram for the pulley:

      The pulley is not accelerating, so we have:

      Thus, using the results of Part B, we have:

      We can then substitute to find:

      Which gives:

      which agrees with the problem statement.

      Note that it is impossible to understand the center of mass acceleration using our "mathematically equivalent" system of Part A Method 2.

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