Question

# An Atwood's machine consists of two masses, m1 and m2, connected by a string that passes...

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.

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

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