An Atwood machine is a classic physics setup with two hanging masses connected by a light string over a pulley. It is useful because it turns Newton's second law into a clear, measurable motion problem. When the masses are unequal, the heavier mass accelerates downward while the lighter mass accelerates upward.
The model helps students connect force diagrams, acceleration, and string tension in one system.
To solve an Atwood machine, draw a free-body diagram for each mass and choose the positive direction along each mass's motion. Gravity pulls each mass downward, while the string tension pulls upward on both masses. If the string is massless and the pulley is frictionless, both masses have the same magnitude of acceleration and the same tension acts throughout the string.
Combining the two Newton's second law equations gives formulas for the acceleration and tension.
Key Facts
- For masses m1 and m2 with m2 > m1, the acceleration magnitude is a = (m2 - m1)g / (m1 + m2).
- The tension can be found from the lighter mass: T - m1g = m1a, so T = m1(g + a).
- The tension can be found from the heavier mass: m2g - T = m2a, so T = m2(g - a).
- Both masses have the same acceleration magnitude because they are connected by the same taut, inextensible string.
- The net external driving force on the two-mass system is (m2 - m1)g when pulley friction and string mass are ignored.
- If m1 = m2, then a = 0 and T = mg for each mass, so the system is in equilibrium if released from rest.
Vocabulary
- Atwood machine
- A system of two masses connected by a string over a pulley, used to study Newton's second law and tension.
- Tension
- The pulling force transmitted through a stretched string, rope, or cable.
- Free-body diagram
- A diagram that shows all external forces acting on one object.
- Acceleration
- The rate at which velocity changes, including changes in speed or direction.
- Net force
- The vector sum of all forces acting on an object or system.
Common Mistakes to Avoid
- Using different acceleration magnitudes for the two masses is wrong because a taut, massless string makes both masses move together with the same magnitude of acceleration.
- Writing the same force equation for both masses is wrong because one mass accelerates upward and the other downward, so the signs must match the chosen positive directions.
- Setting tension equal to weight for a moving mass is wrong because T = mg only when that mass has zero acceleration.
- Forgetting that tension is an internal force for the two-mass system is wrong because it cancels when the two masses are treated as one combined system.
Practice Questions
- 1 An Atwood machine has m1 = 2.0 kg and m2 = 5.0 kg. Using g = 9.8 m/s^2, find the acceleration magnitude and the string tension.
- 2 Two masses, 3.0 kg and 4.0 kg, are connected over a frictionless pulley. Find the acceleration of the system and the tension in the string using g = 9.8 m/s^2.
- 3 If the pulley has friction or the string has noticeable mass, explain why the simple formulas a = (m2 - m1)g / (m1 + m2) and T = constant may no longer be valid.