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.
Understanding Physics: Tension in Ropes and Cables
A rope can only pull, so it cannot provide a pushing force when it goes slack. This is an important first check in any problem. If the rope is loose, its tension is zero and the connected objects may move separately.
When the rope is tight, its length links the motion of the objects. If one end moves one metre upward, another end of the same single rope must move in a matching way set by the pulley arrangement.
This motion rule is often needed before calculating forces. A force diagram alone does not tell the whole story.
Start each connected-system problem by drawing one free-body diagram for every object, not one diagram for the whole setup. Include weight, contact forces, friction where present, and tension from every rope segment attached to that object. Choose a positive direction for each object.
Then write Newton's second law separately. Objects joined by a taut rope usually have the same acceleration size, though their directions can differ.
A hanging mass may accelerate downward while a block on a table accelerates horizontally. The rope creates that shared acceleration, but it does not make the forces on the objects identical.
Pulleys can change the direction of a force or spread a load across several rope sections. A movable pulley attached to a load is supported by more than one upward tension force. For two supporting sections, the upward pull on the pulley can be twice the tension in one section.
This is why a worker can lift a heavy load with a smaller applied force by pulling a longer distance. The trade-off is motion. To raise the load by one metre, the free end may need to move two metres or more.
Energy is not created. Less force is exchanged for more pulling distance.
Real ropes and cables are not ideal. A rope with mass can have greater tension near its top because that part supports the rope below. Friction in a pulley can make the tension different on its two sides.
A stretching rope stores elastic energy and can cause bouncing or vibration. Engineers must consider the largest possible tension, not just the usual value. Elevators, cranes, bridge cables, climbing ropes, and seat belts all need safety margins because loads can change quickly.
When solving school problems, read assumptions carefully. Words such as massless, smooth, light, inextensible, or frictionless tell you which real effects to ignore. If those words are absent, decide whether the problem gives enough information to include a nonideal effect.
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.