Atwood's Machine

Atwood's machine is a device where two masses, M & m, with M > m, are connected by a massless string passing over a massless pulley. What is the acceleration of each mass?

  1. Diagram and coordinate systems
  2. Isolate the systems
  3. Draw all forces acting
  4. Take components
  5. Apply F=ma and apply constraints
  6. Solve
  7. Check!

Steps 1-4: See diagram and transparency.

Step 5: Apply Newton's 2nd law and constraints

Constraints: non-stretchy rope implies T1 = T2 = T   and   aM=-am.

Newton's Second Law: mam = T-mg   and    MaM = T-Mg

Step 6: Solve Newton's Second Law

+MaM = T-Mg

-maM = T-mg

Subtracting the two equations gives: (M + m) aM = (m-M) g
aM =
m - M
M + m

Step 7: Check your answer! Look at simple limiting cases