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Atwood Machine

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, m1 and m2 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, m1, experiences two forces: m1g from gravity, and T, the tension force from the rope. Newton's second law for m1 then states that:

m1a1=m1g+T,

where a1 is the acceleration of m1 in the upwards direction. The second mass, m2, experiences a net force of:

m2a2=m2g+T.

where a2 is the acceleration of m2. Notice that in order for the rope to maintain its total length, the accelerations must be equal and opposite, hence a1= a2. Now subtracting the second equation from the first equation gives:

m1a1m2a2=m1g+m2g.

Finally, making the substitution a1= a2 and rearranging for a1 yields:

a1=gm2m1m2+m1.

 

Note that if m2>m1, the first mass will accelerate upwards and m2 will accelerate downwards, and if m1>m2 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 = y0+a t2, 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.)

 

Adjust the masses of the blocks with the sliders and press the "Animate" button to drop them.

Left Mass

Right Mass

 

 

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