Introduction

Many everyday situations involve ropes, cables, chains, or strings pulling on objects. Whether a crane lifts a heavy load, an elevator carries passengers, or a climber hangs from a rope, the pulling force transmitted through the rope is called tension.

Tension is one of the most common contact forces studied in mechanics. Understanding tension helps us analyze connected objects, draw accurate free-body diagrams, and solve many real-world physics problems.

---

What is Tension?

Tension is the pulling force transmitted through a rope, cable, chain, or string when it is pulled tight by forces acting at its ends.

Unlike a push, tension can only pull.

A rope cannot push an object because it becomes slack instead.

Examples of tension include:

• A crane lifting a steel beam.
• A person pulling a sled with a rope.
• An elevator supported by steel cables.
• A mountain climber hanging from a climbing rope.

---

How Does Tension Arise?

Tension is produced whenever a rope or cable is stretched.

For example:

A person pulls on one end of a rope attached to a box.

• The person pulls on the rope.
• The rope pulls on the box.
• The box pulls back on the rope.

The rope transmits the pulling force from one object to another.

According to Newton's Third Law, each interaction produces equal and opposite forces.

---

Direction of the Tension Force

A very important rule is:

Tension always acts along the length of the rope or cable.

It always pulls away from the object.

For example:

Horizontal Rope

A rope pulls a box to the right.

The tension acts horizontally to the right.

---

Vertical Rope

A hanging object experiences tension upward.

The rope pulls upward on the object.

---

Angled Rope

If a rope is at an angle, the tension acts along the rope at that same angle.

What Is Tension?

Tension is a pulling force transmitted through a rope, string, cable, chain, or similar object when it is pulled tight.

The symbol normally used for tension is:

T

Tension is measured in newtons (N) because it is a force.

A rope can transmit a force from one object to another. For example, when you pull a sled using a rope, the rope pulls on the sled.

Person → Rope → Sled

The force transmitted through the rope is the tension force.

An important rule is:

Tension always pulls. A rope cannot push an object.

If a rope becomes slack, it no longer provides tension.

---

How Is Tension Produced?

Imagine two people pulling on opposite ends of a rope.

Each person pulls on the rope, causing the rope to become stretched slightly. Internal forces within the rope transmit the pulling force from one end to the other.

The same principle applies to:

• cables supporting bridges
• ropes lifting objects
• elevator cables
• tow ropes
• climbing ropes
• strings holding suspended objects
• cables in cranes

Tension therefore allows forces to be transmitted over a distance.

---

The Direction of Tension

Tension acts along the rope or cable.

When drawing a tension force, the arrow points:

away from the object and along the rope.

Consider a mass hanging from a rope.

↑ T

Mass

↓ Fg

The rope pulls upward on the mass, while gravity pulls downward.

This is one of the simplest tension systems.

---

Tension on a Free-Body Diagram

A free-body diagram shows all the forces acting on one object.

When a rope or cable is attached to an object, tension should usually appear as a force pointing along the rope and away from the object.

For a hanging object:

↑ T

●

↓ Fg

For a box being pulled horizontally:

T →

[BOX]

← Friction

↑ N

↓ Fg

Remember that the free-body diagram should show forces acting on the object, not forces that the object exerts on something else.

---

A Hanging Object at Rest

Suppose a 5.0 kg mass hangs from a rope and remains stationary.

Two forces act on the mass:

• tension upward
• gravitational force downward

Because the object is stationary:

a = 0 m/s²

Therefore:

Fnet = 0 N

The forces must balance:

T = Fg

Since:

Fg = mg

then:

T = mg

For the 5.0 kg mass:

T = 5.0 × 9.8

T = 49 N

The rope therefore has a tension of 49 N.

---

Tension Does Not Always Equal Weight

A common mistake is to assume:

T = mg

This is only true in certain situations.

If an object is accelerating vertically, the tension and weight are not balanced.

For example:

↑ T

● accelerating upward

↓ mg

If the object accelerates upward:

T > mg

If the object accelerates downward:

T < mg

If the object has zero acceleration:

T = mg

This gives us an important relationship between tension and motion.

---

Using Newton's Second Law

Tension problems are usually solved using:

Fnet = ma

The first step is to identify the forces and choose a positive direction.

Suppose upward is positive.

For a vertically accelerating object:

T − mg = ma

This equation can be rearranged to find tension:

T = mg + ma

or:

T = m(g + a)

This applies when the object accelerates upward.

---

Worked Example 1: Hanging Mass at Rest

A 12 kg object hangs motionless from a cable.

Calculate the tension.

Because the object is stationary:

Fnet = 0

Therefore:

T = mg

T = 12 × 9.8

T = 117.6 N

Approximately:

T = 118 N

---

Worked Example 2: Accelerating Upward

A 10 kg object is lifted upward with an acceleration of 2.0 m/s².

Calculate the tension in the rope.

The forces are:

↑ T

●

↓ mg

Use:

Fnet = ma

Taking upward as positive:

T − mg = ma

Substitute:

T − (10 × 9.8) = 10 × 2.0

T − 98 = 20

T = 118 N

Tension = 118 N

Notice that the tension is greater than the object's weight because an upward net force is required.

---

Worked Example 3: Accelerating Downward

A 10 kg object is lowered with a downward acceleration of 2.0 m/s².

The forces are still:

↑ T

●

↓ mg

Now the acceleration is downward.

Taking downward as positive:

mg − T = ma

98 − T = 10 × 2.0

98 − T = 20

T = 78 N

Tension = 78 N

The tension is less than the weight because the resultant force must be downward.

---

Tension in a Horizontal System

Tension can also pull objects horizontally.

Consider a 10 kg box being pulled across a frictionless floor by a rope with a tension of 30 N.

Horizontally:

Fnet = T

Using Newton's Second Law:

T = ma

30 = 10a

a = 3.0 m/s²

Therefore:

a = 3.0 m/s²

The tension causes the box to accelerate.

---

Tension with Friction

Real surfaces often produce friction.

Suppose a rope pulls a box to the right while friction acts to the left.

← Friction

[BOX]

Tension →

The net horizontal force is:

Fnet = T − Ffriction

Therefore:

T − Ffriction = ma

---

Worked Example 4: Tension and Friction

A 20 kg box is pulled horizontally by a rope with a tension of 80 N. Friction acts with a force of 30 N.

Calculate the acceleration.

First calculate the net force:

Fnet = T − Ffriction

Fnet = 80 − 30

Fnet = 50 N

Now use:

Fnet = ma

50 = 20a

a = 2.5 m/s²

Acceleration = 2.5 m/s²

---

Two Objects Connected by a Rope

Tension becomes especially useful when analyzing objects connected together.

Consider two boxes connected by a rope:

[5 kg] — rope — [10 kg] → Applied Force

When the system accelerates, the rope transmits force between the two boxes.

One useful strategy is to first treat both boxes as one system.

Suppose a 45 N force pulls the two boxes across a frictionless surface.

Total mass:

m = 5 + 10

m = 15 kg

Calculate acceleration:

F = ma

45 = 15a

a = 3.0 m/s²

Both boxes have the same acceleration because the connecting rope remains taut.

Now we can calculate the tension.

For the 5 kg box:

T = ma

T = 5 × 3

T = 15 N

The tension between the boxes is therefore 15 N.

---

Why Isn't the Tension 45 N?

This is an important question.

The 45 N external force accelerates the entire 15 kg system.

The tension between the boxes only needs to accelerate the 5 kg box in this example.

Therefore:

Applied force = 45 N

but:

Tension = 15 N

Tension should not automatically be assumed to equal the applied force.

---

Ideal Ropes

In introductory physics problems, ropes are often treated as ideal ropes.

An ideal rope is assumed to be:

• massless
• unable to stretch
• perfectly flexible

Under these assumptions, the tension is the same throughout a continuous rope when it passes through ideal conditions.

Real ropes have mass and can stretch, so real tension systems can be more complicated.

---

Tension and Pulleys

Pulleys are commonly used to change the direction of a tension force.

For example, pulling downward on one end of a rope can lift an object upward at the other end.

In an ideal pulley system:

• the rope is massless
• the pulley has no friction
• the tension is the same throughout the rope

This allows us to analyze connected objects using Newton's Second Law.

---

Two Hanging Masses

A classic tension problem involves two masses connected by a rope over a pulley. This arrangement is often called an Atwood machine.

If one mass is heavier than the other, the heavier mass accelerates downward while the lighter mass accelerates upward.

For the heavier mass:

m₂g − T = m₂a

For the lighter mass:

T − m₁g = m₁a

Both objects have the same magnitude of acceleration when connected by an ideal rope.

These equations can be solved together to determine the acceleration and tension.

---

Tension at an Angle

A rope does not always pull horizontally or vertically.

Suppose someone pulls a sled using a rope angled upward.

The tension force can be separated into components:

Horizontal component:

Tx = T cos θ

Vertical component:

Ty = T sin θ

The horizontal component helps move the object forward.

The vertical component pulls upward and can reduce the normal force.

A smaller normal force may also reduce friction because:

Ffriction = μN

This is why pulling something upward at an angle can sometimes make it easier to move.

---

Tension in Everyday Life

Tension forces appear in many everyday structures and machines.

Examples include:

• elevator cables supporting an elevator
• crane cables lifting construction materials
• climbing ropes supporting climbers
• tow ropes pulling vehicles
• suspension bridge cables supporting bridge decks
• strings supporting hanging lights
• cables holding antennas and towers
• ropes used in sailing

Engineers must calculate the tension these cables experience so that they can select materials strong enough to withstand the forces safely.

---

Breaking Tension

Real ropes and cables can only withstand a certain maximum tension.

This is called their breaking strength or maximum allowable tensile force.

If:

T > maximum safe tension

the rope or cable may break.

Engineers therefore use safety factors when designing cables, bridges, elevators, cranes, and other structures.

A cable may be designed to withstand forces much larger than the forces it normally experiences.

---

Did You Know?

Suspension bridges rely heavily on tension. The main cables are pulled tight as they support the weight of the bridge deck. These cables transfer forces through the bridge structure toward the towers and anchorages.

Steel is particularly useful for these applications because steel cables can withstand very large tensile forces.

---

Common Mistakes with Tension

Mistake 1: Drawing tension toward the object

Tension pulls along the rope away from the object.

Mistake 2: Assuming T = mg in every problem

This only applies when the vertical forces balance.

Mistake 3: Assuming tension always equals the applied force

In systems containing multiple objects, the tension may be different from the external applied force.

Mistake 4: Drawing tension when there is no rope, cable, string, or similar connection

Tension only exists when a connecting object is transmitting a pulling force.

Mistake 5: Forgetting other forces

A tension problem may also involve:

• gravity
• normal force
• friction
• drag

Always construct a free-body diagram before choosing an equation.

---

A Strategy for Solving Tension Problems

When solving a tension problem:

1. Identify the object or system being analyzed.
2. Draw a free-body diagram.
3. Identify every force acting on the object.
4. Choose a positive direction.
5. Determine whether the forces are balanced.
6. Write Fnet = ma for each relevant direction.
7. Substitute the known values.
8. Solve for the unknown tension or acceleration.
9. Check whether the answer makes physical sense.

For connected objects, it is often useful to calculate the acceleration of the whole system first and then analyze one object separately to find the tension.

---

Key Terms

Tension: A pulling force transmitted through a rope, cable, string, chain, or similar connector.

Free-body diagram: A diagram showing all external forces acting on an object.

Net force: The vector sum of all forces acting on an object.

Equilibrium: A condition in which the net force is zero.

Ideal rope: A theoretical rope with no mass and no stretching.

Pulley: A wheel used with a rope or cable to change the direction of a force.

Breaking strength: The maximum tension a rope or cable can withstand before failing.

---

Key Equations

Newton's Second Law:

Fnet = ma

Weight:

Fg = mg

Stationary hanging object:

T = mg

Object accelerating upward:

T − mg = ma

Object accelerating downward:

mg − T = ma

Horizontal object with friction:

T − Ffriction = ma

Angled tension:

Tx = T cos θ

Ty = T sin θ

---

Key Takeaways

• Tension is a pulling force transmitted through a rope, cable, string, or similar connector.
• Tension is measured in newtons (N).
• Tension acts along the direction of the rope.
• A tension arrow on a free-body diagram points away from the object along the rope.
• A rope can pull but cannot push.
• For an object hanging at rest, T = mg.
• When an object accelerates upward, tension is greater than its weight.
• When an object accelerates downward, tension is less than its weight.
• Connected objects can be analyzed using Fnet = ma.
• Objects connected by an ideal taut rope have the same magnitude of acceleration.
• Tension should not automatically be assumed to equal the applied force.
• Pulleys can change the direction of tension forces.
• Real ropes and cables have maximum safe tensions that engineers must consider.