The Atwood machine is a fundamental physics demonstration device invented by George Atwood in 1784. It consists of two masses connected by a string that passes over a pulley. This elegant system allows us to study the principles of mechanics in a controlled way. When the masses are unequal, the heavier mass accelerates downward while the lighter mass accelerates upward, creating motion in the system.
To understand how the Atwood machine works, we need to analyze the forces acting on each mass. For each mass, there are exactly two forces: the weight acting downward due to gravity, and the tension in the string acting upward. The weight of mass one is m1 times g, and the weight of mass two is m2 times g. The tension T is the same throughout the string because we assume the string is massless and inextensible.
Now we apply Newton's second law to each mass in the Atwood machine. Newton's second law states that the net force equals mass times acceleration. For mass one, we choose downward as positive, so the equation becomes m1 times g minus T equals m1 times a. For mass two, we choose upward as positive, giving us T minus m2 times g equals m2 times a. This consistent sign convention is crucial for solving the system correctly.
Now we solve for the acceleration by eliminating the tension from our two equations. We add the equations together: m1 g minus T plus T minus m2 g equals m1 a plus m2 a. The tension terms cancel out, leaving us with m1 g minus m2 g equals m1 a plus m2 a. Factoring gives us m1 minus m2 times g equals m1 plus m2 times a. Finally, solving for acceleration, we get a equals m1 minus m2 times g divided by m1 plus m2.
To find the tension, we substitute our acceleration formula back into one of the original equations. Using T minus m2 g equals m2 a, we get T equals m2 g plus m2 a. Substituting our acceleration formula gives us T equals m2 g plus m2 times the fraction (m1 minus m2) g over (m1 plus m2). After factoring and algebraic simplification, we arrive at the final tension formula: T equals 2 m1 m2 g divided by m1 plus m2.