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   a_M=-a_m.

Newton's Second Law: ma_m = T-mg   and    Ma_M = T-Mg

Step 6: Solve Newton's Second Law

+Ma_M = T-Mg

-ma_M = T-mg

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

aM

=

m - M M + m

g

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

• If m = M then a_M = 0. The system is balanced.
• If m = 0 then a_M = -g. M is in freefall.
• If m > M then a_M is positive. M accelerates up.