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Equation for uniform acceleration

Atwood machine.

We are able to derive an equation for the acceleration by using force analysis. If we consider a massless, inelastic string and an ideal massless pulley the only forces we have to consider are: tension force (T), and the weight of the two masses (mg). To find an acceleration we need to consider the forces affecting each individual mass. Using Newton's second law we can derive a system equation's for the acceleration (a). Forces affecting m1:

${\displaystyle \;T-m_{1}g=m_{1}a}$

forces affecting m2:

${\displaystyle \;m_{2}g-T=m_{2}a}$

and now from the first equation

${\displaystyle \;T=m_{1}a+m_{1}g}$

substitute it to the second equation we obtain

${\displaystyle \;m_{2}g-m_{1}a-m_{1}g=m_{2}a}$

${\displaystyle \;m_{2}g-m_{1}g=m_{2}a+m_{1}a}$,

and at last

${\displaystyle a=g{m_{2}-m_{1} \over m_{1}+m_{2}}}$

Conversely, the acceleration due to gravity, g, can be found by timing the movement of the weights, and calculating a value for the uniform acceleration a: ${\displaystyle d={1 \over 2}at^{2}}$.