Tension

Tension and connected particles

Tension is the pulling force transmitted along a stretched string, rope, cable, chain, or similar connector. In elementary particle mechanics the connector is often idealized as a massless, inextensible string. Under that idealization the string constrains the motion of the particles attached to it while transmitting a force directed along the string.

A useful tension model must distinguish the physical force from the assumptions used to simplify it. In particular, a real massive rope can have different tension at different points, and a massive or frictional Pulley can produce different tension on its two sides.

1 Physical meaning of tension

A string under tension is pulled at its ends. A small element of the string is pulled by its neighbors, and the string in turn pulls on any object attached to it. The force exerted by an ideal string at a point of attachment acts along the local direction of the string and away from the attached body.

A flexible string can pull but cannot push. Consequently,

If a calculation predicts a negative value of T, the assumed taut-string configuration is not physically valid; the string would instead go slack and the constraint must be reconsidered.

Tension has the dimensions of force and is measured in newtons in SI units.

For a hanging particle of mass m, taking upward as positive,

The familiar result T = mg is therefore a special equilibrium or constant-velocity case, not a general identity.

2 The ideal string model

The standard introductory model assumes that a string is

1. massless,
2. inextensible,
3. perfectly flexible,
4. taut unless stated otherwise.

When the string passes over a pulley, the usual additional idealizations are that the pulley is massless and frictionless and that the string does not slip.

2.1 Why a massless straight string has uniform tension

Consider a small straight segment of an ideal massless string. Let the tensions at its ends be T₁ and T₂. Newton’s second law gives

For a massless segment, m_seg = 0, so

Thus the tension magnitude is the same throughout one continuous ideal string, provided no non-ideal element intervenes.

This derivation is more informative than memorizing “the tension is the same everywhere.” If the string has appreciable mass, or if a massive pulley must be angularly accelerated, the premise changes and equal tension need not follow.

3 Connected particles and kinematic constraints

Particles joined by an inextensible string do not move independently. The fixed total string length imposes a constraint equation. For the simplest one-to-one connection,

where L is constant. Differentiating,

and differentiating again,

The accelerations therefore have equal magnitudes and opposite signs if the two coordinates are measured in the same positive sense.

A reliable solution procedure is:

1. draw a Free-body diagram for each particle;
2. choose consistent positive directions;
3. write Newton’s second law for each particle;
4. write the string-length constraint;
5. solve the simultaneous equations for acceleration and tension.

4 Two particles on a smooth horizontal surface

Consider masses m₁ and m₂ on a frictionless horizontal surface, joined by a light string, with an external force F applied to m₂.

The inextensible string gives

For m₁,

For m₂,

Adding the equations eliminates the internal tension:

so

This example also illustrates system boundaries. If the two blocks are treated as one system, the tension is internal and cancels. If either block is isolated, tension is an external force on that block.

5 A mass on a table connected to a hanging mass

Let m₁ lie on a frictionless horizontal table and let m₂ hang vertically. The masses are connected by one ideal string over a frictionless massless pulley.

Taking rightward as positive for m₁ and downward as positive for m₂,

Eliminating T gives

6 Atwood machine

An Atwood Machine consists of two masses connected by a light inextensible string passing over a frictionless massless pulley.

Assume m₂ > m₁. Choose upward as positive for m₁ and downward as positive for m₂. Then

Adding,

so

Substitution gives

If m₁ = m₂, then a = 0 and T = mg. If one mass is much larger than the other, the magnitude of the acceleration approaches g.

7 Incline connected to a hanging mass

Let m₁ lie on a frictionless incline of angle 𝜃 and let m₂ hang vertically.

Suppose m₂ tends to move downward, pulling m₁ up the incline. Along the string directions,

Hence

The sign of the numerator determines the actual direction of motion. The tension is

If kinetic friction acts on m₁, the term μ_kN must be added with a sign opposing the relative motion.

8 Several masses connected in series

For three masses on a smooth horizontal surface joined by two different strings, the tensions need not be equal because they belong to different strings.

Let an external force F act on m₃. All three masses have common acceleration

For m₁,

For the subsystem m₁ + m₂,

Thus T₂ > T₁ for positive masses: the connector nearer the pull must accelerate more of the system.

9 Massive ropes and nonuniform tension

Equal tension is not a universal property of ropes. Consider a rope with nonzero mass hanging vertically and supporting a load.

In static equilibrium the tension increases upward because each higher cross section supports the load plus more of the rope’s own Weight. If λ is the linear mass density and y is measured upward from the lower end of a uniform rope,

The massless-string result is recovered in the limit λ → 0.

10 Massive pulley: unequal tensions

If the pulley has moment of inertia I and angular acceleration α, the tensions on its two sides generally differ. For a string that does not slip,

The pulley torque equation is

so

Equal tension is recovered only in the ideal limit I → 0 when other dissipative effects are absent.

11 Tension, tensile stress, and surface tension

The word “tension” is used in several related but distinct ways.

• In particle mechanics, T usually denotes a force transmitted along a string or cable.
• In solid mechanics, tensile stress is force per area:
  
• Surface tension is a different physical quantity with units of force per length.

12 Worked examples

Example 1: elevator cable

A 500 kg elevator accelerates upward at 1.2 m∕s². Then

so

Example 2: two blocks on a table

Let m₁ = 3 kg, m₂ = 5 kg, and F = 24 N. Then

Example 3: table and hanging mass

Let m₁ = 4 kg on the table and m₂ = 2 kg hanging. Then

and

Example 4: Atwood machine

Let m₁ = 3 kg and m₂ = 5 kg. Then

and

Example 5: incline and hanging mass

Let m₁ = 4 kg on a frictionless 30^∘ incline and m ₂ = 3 kg hanging. Then

The tension is

Example 6: three masses in series

Let m₁ = 1 kg, m₂ = 2 kg, m₃ = 3 kg, and F = 18 N. Then

13 Common mistakes

1. Assuming T = mg for every hanging object.
2. Assuming every rope in a problem has the same tension.
3. Forgetting that equal tension applies only along one continuous ideal string.
4. Giving two particles different acceleration magnitudes when an inextensible one-to-one constraint requires equal magnitudes.
5. Using inconsistent positive directions on the two sides of a pulley.
6. Forgetting that a flexible string cannot push; T < 0 signals a failed taut-string model.
7. Ignoring pulley inertia when the pulley is explicitly massive.

14 Practice exercises

1. A 6 kg mass hangs at rest from a vertical rope. Find the tension.
2. The same mass accelerates upward at 2 m∕s². Find the tension.
3. The same mass accelerates downward at 2 m∕s². Find the tension.
4. Two blocks of 2 and 3 kg are pulled on a smooth table by 20 N. Find the acceleration and the tension.
5. A 5 kg block on a smooth table is connected to a hanging 2 kg mass. Find a and T.
6. Solve the ideal Atwood machine for m₁ = 4 kg and m₂ = 7 kg.
7. Derive the general Atwood acceleration without first solving for T.
8. A 4 kg block lies on a smooth 25^∘ incline and is connected to a 3 kg hanging mass. Determine the direction of motion.
9. Repeat the preceding problem if μ_k = 0.15.
10. Three masses in series have masses 1, 2, 4 kg and are pulled by 28 N. Find both string tensions.
11. A uniform rope of length L and mass M hangs vertically supporting a mass m. Find the tension at the bottom and top of the rope.
12. Show that a massless pulley implies equal tension if bearing Friction is neglected.
13. For a massive pulley with I = MR², derive T ₂ − T₁ in terms of M and a.
14. Explain physically what must happen if an algebraic solution gives T < 0.

15 GRE-style speed checks

1. A mass m hangs motionless from a massless string. The tension is (A) 0 (B) mg∕2 (C) mg (D) 2mg. Answer: C.
2. A mass m accelerates downward with magnitude a. The tension is (A) m(g + a) (B) m(g − a) (C) ma (D) mg. Answer: B.
3. Two blocks on a smooth table are connected by a light string. The string guarantees that the two blocks have (A) equal forces (B) equal masses (C) equal acceleration magnitude (D) equal momentum. Answer: C.
4. In an ideal Atwood machine with m₂ > m₁, the tension satisfies (A) T > m₂g (B) T = m₂g (C) m₁g < T < m₂g (D) T < m₁g. Answer: C.
5. A continuous ideal string passes over a massless frictionless pulley. The two tension magnitudes are (A) always different (B) equal (C) zero (D) proportional to pulley radius. Answer: B.
6. If a calculated string tension is negative, the best interpretation is (A) compression in the string (B) reversed gravity (C) the assumed taut-string model fails (D) negative mass. Answer: C.
7. A massive pulley accelerates angularly. In general the two string tensions are (A) equal (B) unequal (C) both zero (D) independent of angular acceleration. Answer: B.
8. For several masses connected in series and pulled from one end, the tension nearest the pull is generally larger because it must accelerate (A) less mass (B) more of the connected system (C) only the rope (D) gravity. Answer: B.

16 Source and licensing note

The opening definition and discussion of one-dimensional tension were informed by the Wikipedia article Tension (physics), available under the Creative Commons Attribution–ShareAlike license.

The connected-particle development, constraint treatment, worked examples, problem sequence, and figures in this entry are newly written for PhysicsLibrary. Standard Newtonian results were cross-checked against openly licensed LibreTexts material.

References

Unless otherwise noted, this PhysicsLibrary entry is intended for release under the Creative Commons Attribution–ShareAlike 4.0 International license.