Atwood\'s Machine An Atwood\'s machine consists of two masses, and , connected b

Question

Atwood's Machine An Atwood's machine consists of two masses, and , connected by a string that passes over a pulley. If the pulley is a disk of radius R and mass M find the acceleration of the masses. Express your answer in terms of the variables m1, m2, R, M, and appropriate constants.

Explanation / Answer

?F = m1a = T1 - m1g---->(1) ?F = m2a = m2g - T2----->(2) For the pulley the rotational analog of Newton's 2nd law ?t = Ia = RF Since angular acceleration (a) equals a / R, and the force F acting on the pulley is the two tensions, this becomes: Ia / R = R(T2 - T1) Ia / R² = T2 - T1--->(3) Add (1), (2), and (3) together: m1a + m2+ Ia / R² = m2g - m1g Solved for a: a = g(m2 - m1) / (m1+ m2+ I / R² ) moment of inertia (I),assume a solid pulley (cylinder), is 0.5MR² a = g(m2 - m1) / (m1+ m2+ 0.5MR² / R² ) = g(m2 - m1) / (m1+ m2+ 0.5M)

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