Two masses, M and m, are connected by a string passing over a pulley. Assume that M > m. The pulley is a solid disk of mass mp and radius r. What is the acceleration of the two masses?
Take +y up for mass M and mass m.
Take into plane (clockwise) to be positive for the pulley.
|For mass M:|||||For mass m:|||||For the pulley:|
|ΣFy = Ma1y|||||ΣFy = ma2y|||||Στ = I α|
|T1 - Mg = Ma1|||||T2 - mg = ma2|||||rT1 - rT2 = ½ mpr2α|
5 unknowns, 3 equations of motion, and two constraints:
a1 = - a2 ≡ a α = a /r
The pulley equation then becomes: T1 - T2 =½ mp a
Combining the three equations to eliminate the two tensions gives:
(Mg - Ma) - (mg + ma) =½ mp a
Note: if mp = 0, then a = g (M - m) /
(M + m).