The formula
- Hanging mass at rest or moving at constant speed
T = mg- Hanging mass accelerating vertically
T = m(g + a)- Two masses over a pulley (Atwood machine)
a = (m₂ - m₁)g / (m₁ + m₂) and T = 2m₁m₂g / (m₁ + m₂)- General case, applied to each object separately
ΣF = ma
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
T | Tension in the rope | N |
m | Mass hanging from the rope | kg |
g | Acceleration due to gravity | 9.80 m/s² |
a | Acceleration of the mass, positive upward in the vertical case | m/s² |
m₁, m₂ | The two masses on either side of the pulley, with m₂ the heavier one | kg |
ΣF | Net force on the object being analyzed | N |
When it applies
- The rope is treated as massless and inextensible and the pulley as frictionless. A real rope with mass carries a tension that varies along its length.
- Draw a free-body diagram for each object and apply ΣF = ma to each separately. The shared tension is what links the two equations.
- Objects joined by a taut rope share the same magnitude of acceleration.
Worked example
Problem. A 15 kg crate hangs from a rope inside an elevator that is accelerating upward at 2.0 m/s². Find the tension, and compare it with the elevator at rest.
- Forces on the crate: tension T upward, weight mg downward. Take up as positive.
- Newton's second law along the vertical: T - mg = ma, so T = m(g + a).
- Substitute: T = (15 kg)(9.8 + 2.0) = (15)(11.8) = 177 N.
- At rest or at constant speed, a = 0 and T = mg = (15)(9.8) = 147 N, so accelerating upward adds 30 N.
Answer. 177 N while accelerating upward, against 147 N when the elevator is at rest.
Common mistakes
- Assuming the tension always equals the weight. That holds only when the acceleration is zero.
- Giving an ideal pulley a different tension on each side.
- Forgetting that the rope pulls on both objects it connects, in opposite directions.
- Adding the two weights on an Atwood machine to get the tension. Solve the two Newton's second law equations together instead.
Related formulas
- Torque formula:
τ = rF·sin θ - Centripetal force formula:
F_c = mv² / r - Newton's second law:
ΣF = ma