Tension is the pulling force carried by a rope, cable, string, or chain. This cheat sheet helps students solve common tension problems by organizing forces before substituting numbers. It is useful for worked examples involving hanging masses, boxes pulled by ropes, angled cables, and simple pulley systems.
Clear diagrams and force equations are the key to avoiding sign and direction errors.
Key Facts
- Tension always pulls along the rope or cable, so a rope attached to an object exerts a force away from the object along the rope.
- For an object in equilibrium, the net force is zero, so and .
- For an object with acceleration, Newton's second law applies in each direction: and .
- For a hanging mass at rest, the tension equals the weight: .
- For a hanging mass accelerating upward, the tension is , so .
- For a hanging mass accelerating downward, the force equation is , so .
- For an angled rope with angle measured from the horizontal, the components are and .
- For an ideal massless rope over a frictionless pulley, the tension is the same everywhere in the same continuous rope.
Vocabulary
- Tension
- Tension is the pulling force transmitted through a stretched rope, cable, string, or chain.
- Free-body diagram
- A free-body diagram is a simplified drawing that shows all external forces acting on one object.
- Equilibrium
- Equilibrium means the net force on an object is zero, so the object is either at rest or moving with constant velocity.
- Weight
- Weight is the gravitational force on an object, calculated by .
- Force component
- A force component is the part of a force acting along a chosen axis, such as or .
- Ideal pulley
- An ideal pulley is massless and frictionless, so it changes the direction of a rope's tension without changing its magnitude.
Common Mistakes to Avoid
- Drawing tension in the wrong direction is incorrect because tension always pulls on an object along the rope and away from the object.
- Using for every hanging mass is wrong because only applies when the mass has zero vertical acceleration.
- Forgetting to split angled tension into components causes incorrect force equations because and affect different directions.
- Mixing up sine and cosine for angled ropes gives wrong components because the adjacent side uses cosine and the opposite side uses sine relative to the chosen angle.
- Assuming both sides of every pulley system have different tensions is wrong for an ideal continuous rope because the same rope has the same tension throughout.
Practice Questions
- 1 A mass hangs at rest from a vertical rope. Find the tension using .
- 2 A object is lifted upward with acceleration . Find the rope tension using .
- 3 A rope pulls a box with tension at above the horizontal. Find and .
- 4 A crate is held by two angled cables instead of one vertical cable. Explain why the tension in each cable can be greater than half the crate's weight.
Understanding Tension in Ropes and Cables Worked Examples
The most important decision happens before any calculation. Choose one object, or one connected group of objects, as the system you will study. Then draw only the forces acting on that system from outside it.
A rope force is external when the rope is outside your chosen system. If two blocks are treated as one system, the pull between them becomes an internal force and does not belong on the combined diagram.
This choice can make a long pulley problem much simpler. It also prevents the common mistake of drawing forces that belong to a different object.
A free body diagram should show forces, not motion. An object moving upward may have a downward net force if it is slowing down. An object moving downward may have an upward net force if it is slowing down.
The acceleration direction, rather than the motion direction, tells you which side of the force equation is larger. Pick positive directions at the start and keep them throughout the problem. If upward is positive, a downward force gets a negative sign.
A negative answer does not mean the work failed. It often means the actual direction is opposite to the direction first chosen.
Angled cables require careful geometry because each cable must provide part of the needed support. A shallow cable has a small vertical part of its pull. It must therefore carry a much larger total pull to hold up the same load.
This is why suspension bridges use tall towers and why a clothesline becomes harder to keep level when a heavy object hangs near its center. In a two cable support, first identify whether the setup is symmetric. Equal angles and equal loads on each side produce equal tensions.
If the angles differ, solve the horizontal balance and vertical balance together. Draw the angle in the same position as it appears in the diagram, then check whether you used the horizontal or vertical component correctly.
Pulley questions often hide the link between the motion of different objects. A single taut rope has a fixed total length. When one end moves a certain distance, another connected end must move in a matching way, though their directions may differ.
More complicated pulley arrangements can trade distance for force. A load supported by several rope sections can need less force from the person pulling, but the person must pull more rope. Real ropes and pulleys are not ideal.
Rope mass, friction, stretching, and bending around a pulley can make the pulls at different points unequal. In school problems, use the stated ideal assumptions, then check that your result fits common sense. A supporting rope should not predict a negative pull, and a rope cannot push an object.