Deriving atwood's gravity formual

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SUMMARY

Atwood derived his gravity formula, g = a(M1 + M2)/(M1 - M2), using Newton's second law, F = ma, and the gravitational force equations F1 = M1g and F2 = M2g. The total force acting on the system is the difference between the gravitational forces on the two masses, leading to the equation F = g(M1 - M2). By equating the forces acting on the system, Atwood demonstrated that the acceleration a can be expressed in terms of gravity g and the masses involved.

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  • Concept of net force in a two-mass system
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I have always been curios to how Atwood derived his formula a=g(m2+m1)/(m2-m1) to get gravity. Does anyone know how he did it? I have read that he used Netwons f=ma but I personally could not find a way to do this using Newtons f=ma.

Thanks.
 
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The formula is:

g = a(M1 + M2)/(M1- M2)

using F = ma and F = mg

Gravity is acting on both the weights therefore:

F1 = M1g

F2 = M2g

The total force acting down is given by:

F = F1 - F2

which is:

F = g(M1 - M2)

As the force is pulling both weights we can view the two weights as one single weight being acted on with a mass of M1 + M2, so using F = ma

F = a(M1 + M2)

We,ve now got two terms for the same force so we can subsitue in for F:

a(M1 + M2) = g(M1 - M2)


changing this around:

g = a(M1 + M2)/(M1- M2)

Q.E.D.
 
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