Atwood Machine - Maple Help

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, ${m}_{1}$ 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 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 . 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  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 , 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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