Atwood's Machine

A block of mass M and a block of mass m are connected by a cord that passes over a frictionless pulley. This is known as an Atwood’s machine. The heavier block falls to the ground. What is the acceleration of the blocks?
 
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The free-body diagrams of the two blocks are shown above. T is the tension in the cord.

We can write the following equations:


Ma = Mg − T


ma = T − mg


Adding these two equations together,


Ma + ma = Mg − mg


(M + m)a = (M − m)g

a = (M − m)g / (M + m)


 

My Way
 
The gravitational potential energy of a body that has dropped the distance y is:


GPE = GPEi − mgy


GPEi is the initial gravitational potential energy of the body.
 
 
 









In the above figure, the heavier block has dropped the distance y and the lighter block has risen the distance y.

The total gravitational potential energy of the two blocks is:


GPE = GPEi + mgy − Mgy


The total kinetic energy of the two blocks is:

KE = ½ Mv ² + ½ mv ²


The derivative of gravitational potential energy is:


DGPE = mg − Mg


The derivative of kinetic energy is:


DKE = Ma + ma


The total energy of the two blocks remains the same as the blocks move. The rate of change of energy is zero. We have:


DGPE + DKE = 0


mg − Mg + Ma + ma = 0


Ma + ma = Mg − mg


(M + m)a = (M − m)g


a = (M − m)g / (M + m)