Main Concept

The Atwood machine is a simple device which was invented by Rev. George Atwood in 1784 to illustrate the dynamics of Newton's laws. It consists of a massless, inextensible string which connects two masses, $m_1$ and $m_2$ through an ideal pulley. (An ideal pulley is one which is assumed to have negligible mass and no friction between itself and the string).  A straightforward application of Newton's laws can predict the acceleration of the blocks and the time it takes for them to reach the ground.

The mass on the left, $m_1$, experiences two forces: $-m_{1}g$ from gravity, and $T$, the tension force from the rope. Newton's second law for $m_1$ then states that:

$m_1{a}_1\=-m_{1}g\+T\,$

where $a_1$is the acceleration of $m_1$ in the upwards direction. The second mass, $m_2$, experiences a net force of:

$m_2{a}_2\=-m_{2}g\+T\.$

where $a_2$ is the acceleration of $m_2$. Notice that in order for the rope to maintain its total length, the accelerations must be equal and opposite, hence $a_1\= -a_2$. Now subtracting the second equation from the first equation gives:

$m_1{a}_1-m_2{a}_2\=-m_{1}g\+m_{2}g\.$

Finally, making the substitution $a_1\= -a_2$ and rearranging for $a_1$ yields:

$a_1\=g\cdot \frac{m_2-m_1}{m_2\+m_1}$.

Note that if $m_2\>m_1$, the first mass will accelerate upwards and $m_2$ will accelerate downwards, and if $m_1\>m_2$ the opposite will happen. In an experiment, you could adjust the masses and measure the time it takes for a mass to reach the ground. The time is given by the solution to the equation $0 \= y_0\+a t^2$, and this provides a way to determine $a$ and ultimately find $g$. By choosing blocks with very similar masses, the acceleration is much slower and so air resistance is less important, thereby giving a more accurate method of computing $g$. (On the other hand, the assumption of a friction-free rope becomes untenable if the masses are too similar.)

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