Atwood's Machine Revisited

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?

Use DID TASC
  1. Diagram and coordinate system
  2. Isolate the system
  3. Draw all forces acting
  4. Take components
  5. Apply F=ma   and/or   τ = I α   origin dependent!  and constraints
  6. Solve
  7. Check!

Step 1:
Take +y up for mass M and mass m.
Take into plane (clockwise) to be positive for the pulley.

Step 5:
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
a =
g (M - m)
M + m + ½ mp

Note: if mp = 0, then a = g (M - m) / (M + m).