Tension is the pulling force carried by a stretched rope, cable, string, or chain. It matters because many physics problems involve objects connected by ropes, such as elevators, pulleys, cranes, and hanging signs. In an ideal rope, the rope is massless and does not stretch, so it transmits the same tension along its length.
This lets us replace the rope with force arrows in a free-body diagram and apply Newton's laws to each object.
Key Facts
- Tension always pulls along the rope and away from the object it acts on.
- For an ideal massless rope over a frictionless pulley, the tension is the same everywhere in the rope.
- Newton's second law for each object is ΣF = ma.
- For an Atwood machine with masses m1 and m2, a = (m2 - m1)g / (m1 + m2) if m2 is heavier.
- For an ideal Atwood machine, T = 2m1m2g / (m1 + m2).
- If an object hangs at rest from one vertical rope, T = mg.
Vocabulary
- Tension
- Tension is the pulling force transmitted through a rope, string, cable, or chain when it is stretched.
- Ideal rope
- An ideal rope is massless, does not stretch, and carries the same tension at every point.
- Frictionless pulley
- A frictionless pulley changes the direction of a rope's tension without reducing or increasing its magnitude.
- Free-body diagram
- A free-body diagram shows all external forces acting on one chosen object using labeled arrows.
- Atwood machine
- An Atwood machine is a system of two masses connected by a rope over a pulley, used to study tension and acceleration.
Common Mistakes to Avoid
- Treating tension as a push is wrong because tension can only pull along the rope, never push an object.
- Assuming tension always equals weight is wrong because T = mg only for a hanging object at rest or moving at constant velocity with one vertical rope.
- Using one force equation for the whole system and also including internal tension is wrong because tension between connected objects cancels when the whole system is treated as one object.
- Forgetting that connected objects share the same acceleration in an ideal rope is wrong because the rope length is fixed, so both masses must speed up together in linked motion.
Practice Questions
- 1 Two masses, m1 = 2.0 kg and m2 = 5.0 kg, are connected by an ideal rope over a frictionless pulley. Find the acceleration of the system and the tension in the rope. Use g = 9.8 m/s^2.
- 2 A 12 kg sign hangs at rest from two identical vertical cables that share the load equally. What is the tension in each cable? Use g = 9.8 m/s^2.
- 3 A rope passes over a frictionless pulley connecting two equal masses. Explain why the system does not accelerate and how the tension compares with each mass's weight.