Question

An Atwood's machine consists of two masses, m1 and m2, 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.

Answer 1

now that the pulley has mass, we can no longer treat the tension in the string as the same everywhere; so call T1 the tension of the string supporting m1, and T2 is the tension in the string supporting m2

assume m1<m2 so that m1 accelerates up and m2 accelerates down (this will be important for our signs)

apply newton's second law to m1:

T1- m1 g = m1 a (eq. 1)

similarly to m2:

T2 - m2 g = - m2 a (eq. 2)

now we consider torques on the pulley

the force on the pulley will be T2-T1; it acts a distance R from the center of the pulley, so generates a torque of (T2-T1)R

this torque = I alpha of the pulley; where I is the moment of inertia and alpha the angular acceleration

treating the pulley as a disk, it has moment of inertia = 1/2MR^2;
angular accel is related to linear accel via a= R alpha or
alpha =a/R;

combining all these gives us

(T2-T1)R=1/2 MR^2(a/R) =>
T2-T1=1/2 Ma (eq.3)

now, subtract eq. 1 from eq. 2:

T2-T1 -m2g+m1g=-m2a-m1a

we know from eq. 3 that T2-T1=1/2Ma, substitute this into the eq directly above, and a little algebra will yield our final result:

a=g(m2-m1)/(1/2 M +m1+m2)

now, notice that if M->0, the result becomes the familar result for an Atwood's machine ignoring the mass of the pulley