KAHIBARO
Table of Contents
Force Through a Rope or String
Tension is the pulling force transmitted through a stretched rope, string, cable, or similar connector. It appears whenever two objects pull on each other through something flexible. A rope can pull, but it cannot push effectively in the usual idealized physics problems.
If you pull one end of a rope, the rope pulls back on your hand, and it also pulls on whatever is attached to the other end. That transmitted pull is called tension.
What Tension Means Physically
A rope is made of matter, so when it is pulled, its particles exert forces on neighboring particles. This internal pulling allows a force applied at one end to be transmitted to the other end. In beginner mechanics, we usually simplify the rope as ideal, meaning it has negligible mass and does not stretch.
In that ideal case, the tension is the same everywhere along a single straight rope segment.
For an ideal rope or string, with negligible mass and no stretching, the tension has the same magnitude throughout the rope.
Tension is usually represented by the symbol $T$.
Direction of Tension
Tension always acts along the rope or string. It pulls away from the object to which the rope is attached.
This is one of the most important ideas. A rope never pushes an object along its length in these basic models. It only pulls.
Suppose a block is attached to a rope on its right side. Then the rope pulls the block toward the right. If the same rope is attached to another object on its left side, then the rope pulls that object toward the left. The tension force on each object points along the rope, away from the object.
The rope pulls object $A$ toward object $B$, and pulls object $B$ toward object $A$.
Tension in Free-Body Diagrams
When drawing a free-body diagram, tension is shown as a force arrow along the rope, pointing away from the object. If an object is attached to more than one rope, each rope can exert its own tension force.
For example, if a mass hangs vertically from a string, the string pulls upward on the mass with tension $T$.
Here the two main forces on the mass are the upward tension $T$ and the downward weight $mg$.
Tension and Equilibrium
If a hanging object is at rest, its acceleration is zero. Then the net force must be zero, so the upward tension balances the downward weight.
$$T = mg$$
This is true only in that specific situation, a single hanging mass at rest or moving at constant velocity.
Tension is not always equal to weight. The relation $T = mg$ is true only when the object has zero vertical acceleration in that simple setup.
If the object accelerates upward, then the tension must be greater than its weight.
$$T - mg = ma$$
So,
$$T = mg + ma$$
If the object accelerates downward with magnitude $a$, then
$$mg - T = ma$$
so
$$T = mg - ma$$
Tension in Horizontal Pulling
Consider a block on a frictionless horizontal surface pulled by a rope. The tension is the horizontal force causing the acceleration.
If the only horizontal force is tension, then Newton's second law gives
$$T = ma$$
In this case, tension is not related to weight directly. The weight and normal force act vertically and cancel each other, while the tension acts horizontally.
Tension in Connected Objects
Tension becomes especially important when two or more objects are connected by a rope. The rope transmits force from one object to another, allowing them to accelerate together.
Suppose two blocks of masses $m_1$ and $m_2$ are connected by a light rope on a frictionless surface, and a force pulls the system.
The rope pulls on each block. Even though the tension has the same magnitude in an ideal rope, it acts in opposite directions on the two different objects.
This is why free-body diagrams must be drawn for each object separately.
Ideal Rope and Real Rope
In many introductory problems, we assume an ideal rope or string. This simplifies the analysis.
| Rope model | Mass | Stretching | Tension along rope |
|---|---|---|---|
| Ideal rope | Negligible | None | Same everywhere |
| Real rope | May matter | May stretch | Can vary from point to point |
If the rope has significant mass, different parts of the rope may need different forces to accelerate, so the tension can change along its length. That more advanced case is usually not treated in simple Newton's laws problems.
Tension and Pulleys
When a rope passes over an ideal pulley, the pulley changes the direction of the tension force, but not its magnitude.
For an ideal rope and ideal pulley, the tension magnitude stays the same on both sides of the pulley, although the direction can change.
This fact is very useful in many mechanics problems. A hanging mass can create tension in a rope, and that same tension can pull another object in a different direction.
Tension Compared with Other Forces
Tension is different from some other common forces in useful ways.
| Force | Acts through | Typical direction |
|---|---|---|
| Tension | Rope, string, cable | Along the rope, pulling |
| Normal force | Surface contact | Perpendicular to surface |
| Friction | Surface contact | Along surface |
| Weight | Gravity | Toward Earth, or toward attracting body |
Tension depends on the interaction through the connector. It is not a property stored in the object alone.
Common Mistakes
A very common mistake is to assume that tension always equals $mg$. That is not generally true. Tension depends on the full motion and force balance of the system.
Another common mistake is to draw tension in the wrong direction. Remember that the rope pulls the object away from the point of attachment, along the rope.
Students also sometimes forget that the same rope can pull different objects in opposite directions. The magnitude may be the same, but the force vectors act on different bodies.
Always draw tension along the rope and away from the object being analyzed.
Simple Example
Imagine a $2\,\text{kg}$ mass hanging at rest from a light string. The forces are tension upward and weight downward.
Using $g = 9.8\,\text{m/s}^2$,
$$mg = 2 \times 9.8 = 19.6\,\text{N}$$
Since the mass is at rest,
$$T = mg = 19.6\,\text{N}$$
Now imagine the same mass accelerates upward at $1.2\,\text{m/s}^2$. Then
$$T = mg + ma$$
$$T = 19.6 + 2(1.2) = 22.0\,\text{N}$$
So the tension is larger than the weight.
Key Idea to Remember
Tension is the pulling force transmitted by a stretched rope, string, or cable. In ideal problems, it acts along the rope and has the same magnitude throughout a single rope segment.
Key rule: tension is a pulling force, directed along the rope, away from the object. For an ideal rope, its magnitude is the same throughout the rope.