Unit 3 · Topic 04
Tension
Tension models the force transmitted by a stretched rope, string, or cable. Its value depends on the system and on the idealizations used for ropes and pulleys.
Unit 3TensionOpen navigation
Concept 01
What tension means
Essential The minimum you should retain
A taut rope pulls bodies attached to its ends along the local direction of the rope. An ideal rope does not push.
UnderstandInterpret and connect
In a body's FBD, tension points away from the body along the rope. The third-law partner acts on the rope and belongs to a different system.
DeepenFormulation and conditions
Tension can be interpreted as the internal force transmitted across an ideal rope section. For a massless rope with no pulley dynamics, a finite tension difference would imply an unbounded acceleration of an ideal massless element; the model therefore gives uniform tension along a connected segment.
ExploreConnections for further study
In a massive rope, tension may vary with position because different sections must accelerate different amounts of mass. «Same tension» is an idealization, not a universal law.
Concept 02
Ideal rope and pulley
Essential The minimum you should retain
For an inextensible massless rope over an ideal pulley, tension has the same magnitude on both sides of the same rope.
UnderstandInterpret and connect
The ideal pulley changes the direction of tension without loss or a magnitude difference. Constant rope length also constrains the connected bodies' motion.
DeepenFormulation and conditions
For two masses connected over an ideal fixed pulley, write two Newton equations plus the kinematic constraint. The common acceleration and tension come from solving the system; T = mg is not assumed.
ExploreConnections for further study
A pulley with rotational inertia generally requires different tensions on its two sides to produce torque. That belongs to rotational dynamics, not the ideal pulley model used here.
Worked example
Ideal Atwood machine
Two masses are connected by an ideal rope over an ideal pulley; m1 = 2 kg and m2 = 3 kg.
- m1 = 2 kg
- m2 = 3 kg
- g = 9.8 m/s²
Find the acceleration magnitude and tension.
- Conventions
Take upward as positive for m1 and downward as positive for m2.
- Equations
m2g - T = m2a and T - m1g = m1a.
- Acceleration
Adding gives (m2-m1)g = (m1+m2)a, so a = 1.96 m/s².
- Tension
T = m1(g+a) = 2(11.76) = 23.52 N.
The ideal rope transmits the same T magnitude on both sides and constrains equal acceleration magnitudes; the weights belong to different systems.
Concept 03
Tension and geometry
Essential The minimum you should retain
When several ropes support a body, each tension points along its rope. Their components combine in the body's force balance.
UnderstandInterpret and connect
Two symmetric ropes can cancel their horizontal components while their vertical components support a load.
DeepenFormulation and conditions
If two equal tensions T make angle θ above the horizontal and support a mass in equilibrium, 2T sinθ = mg. As θ becomes small, T must become very large to keep the same vertical component.
ExploreConnections for further study
A nearly horizontal cable can support a load only with very large tension. This explains why cable geometry and anchor design matter in real structures.
Tensions follow the cables. Their horizontal components cancel and their vertical components support the weight, without revealing T numerically.
Concept review
Common errors
Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.
Assuming T=mg for every vertical rope.
T=mg holds only under particular conditions such as equilibrium of that body.
Extending «same T» to a massive rope or a pulley with inertia.
Equal tension belongs to the ideal model.