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Notebooks · posts/a-box-hanging-from-two-ropes

The tension in each of two ropes holding up a box

Adam Zettel · September 24, 2026 · written with an agent

A box of weight W hangs from two ropes that make angles θ₁ and θ₂ with the ceiling. Balancing the sideways pulls and the upward pulls gives the tension in each rope in terms of W and the angles. The surprise is how large it gets: at 30° each rope holds half the weight, 49 N, but its tension is 98 N, because only half of the tension points up; a nearly flat rope needs a tension many times the weight.

Start with a guess. A box weighing (that is a box, since and ) hangs from two ropes, each making a angle with the ceiling: the angles and marked in the first figure. With two ropes holding it up, is the tension in each rope half the weight, ? Each rope does hold half the weight, , through the upward part of its tension, . But the tension in each rope is : at only half of the tension points up. The other part points sideways and is canceled by the other rope. The tension can be larger than the weight itself. The cells below show why, give the tension for any pair of angles, and hang the same box a second way, with one rope steeper than the other.

Look inside this program

  • Atlas

    The program’s map: every function and which calls which.

  • There’s no recorded run of this program yet.

Shift+Enter in a cell runs it and moves to the next. Running a cell runs every cell above it too, as one program.

The box's weight

A box's weight is the force gravity pulls it down with: its mass times , measured in newtons (N). The program measures angles in degrees, the way problems state them, and converts to radians only where it calls sin and cos.

[1]lines 32–42

A box weighs (weight).

The setup, with its angles

The figures on this page are drawn by the program from the angles themselves. Each angle is measured at the ceiling, between the ceiling and the rope, and marked there with an arc. The function below draws one hanging box: the ceiling, the two ropes from where they are fixed down to the knot, the box under the knot, and the two angles.

[2]lines 55–123

What pulls on the knot

The knot is the point where the two ropes meet, just above the box. If we look at the knot, there are three forces acting on it, and they must cancel, because the knot is not moving. The box pulls straight down with its weight . The left rope pulls up and to the left along its own length with some force , and the right rope pulls up and to the right with . A rope can only pull along itself; that pull is what tension means.

Splitting each pull into sideways and upward parts

The angle a rope makes with the ceiling is also the angle it makes with the horizontal line through the knot: the ceiling and that line are parallel, and the rope crosses both. So a rope at angle pulls on the knot partly sideways and partly up. With tension , the sideways part is and the upward part is : the rope's pull is the long side of a right triangle, gives the side along the horizontal, the side going up. The free-body diagram further down draws those triangles.

Now require that nothing moves.

Sideways: the left rope's sideways pull must cancel the right rope's:

Up and down: the two upward parts together must hold up the weight:

Two equations, two unknowns, and . The function below is those two equations written as code. Given candidate tensions, it adds up the three forces and reports how much is left over. For the right tensions the answer is zero; the checks further down use it on every answer.

[3]lines 155–164

Solving the symmetric case first

When both ropes make the same angle , the sideways equation says . Call it . The up-and-down equation becomes , so

Before running: what is with the ropes straight up (), and at ?

[4]lines 176–185

Straight up, each rope carries half the weight, (ratio_straight_up), as the guess at the top expected. At it is (ratio_at_30_degrees): , so , the answer from the start, now explained. Each rope still holds half the weight through its upward part, . At only half of the tension points up, so the tension is twice that half: , equal to the weight of the box. The rest of the tension, , points sideways and is canceled by the other rope.

Solving for any two angles

From the sideways equation, . Put that into the up-and-down equation, multiply through by , and use :

Each rope's tension has the other rope's angle on top.

[5]lines 205–220

At and the general formula gives (tension_at_30_degrees), the same as the symmetric case. Of each rope's , points sideways (sideways_part_at_30) and points up (upward_part_at_30). The up is the half of the weight each rope holds; the sideways is canceled by the other rope's sideways part.

The free-body diagram, to scale

The next figure is the knot alone, with the three forces on it drawn as arrows, all to one scale. Each rope's pull is split into its sideways part and its upward part, drawn dashed as the two short sides of a triangle. The two sideways parts are equal and opposite; the two upward parts add up to the weight. The rope arrows come out as long as the weight arrow, because each tension is , while each upward part is .

[6]lines 238–276

Two ways to hang the same box

Now hang the same box with the left rope steeper, at , and the right rope still at . Before running: which rope carries more tension? And the rope, which carried in the symmetric case: more or less now?

[7]lines 286–324

With and the steep rope's tension is and the tension in the rope at the smaller angle is (steep_and_shallow): per unit of weight, and (steep_and_shallow_ratio). The steeper rope, the one closer to vertical, carries more tension. The rope is at the same angle as before, yet its tension fell from to , because the steep rope now holds most of the weight: of it, against . The right panel balances the same way the symmetric one does: sideways, , and up, . The two tensions add up to more than , because neither rope points straight up.

[8]lines 338–343

Checking the algebra a second way

The formulas came from a few lines of algebra, and algebra can go wrong. So the program also finds the tensions without them: the two balance equations are a pair of linear equations in and , and lin_solve solves such a system directly. The tests compare the two answers over eight pairs of angles, and check with leftover_force that the formula's tensions cancel the weight.

[9]lines 357–391

Why a flatter rope carries more tension

Before running: was at . At , is it more or less than ? At ?

[10]lines 400–414

More: at (ratio_at_10_degrees) and at (ratio_at_5_degrees). The figure shows the whole curve. At each tension is , and as the ropes flatten out the tension climbs, slowly at first and then steeply. A rope pulled perfectly flat would need infinite tension, because none of its pull would point up. That is why a clothesline sags: no rope can be pulled tight enough to be straight while holding anything.

The angle trap

Problems measure the angle in two ways, and mixing them up is a common mistake. If the angle is measured from the vertical instead of the ceiling, then , and becomes : The ropes at above the ceiling are at from the vertical. Put into the new formula, that should give the same .

[11]lines 433–438

It does: (tension_from_the_vertical). Putting into the cosine formula by mistake would give instead (wrong_angle_ratio), which is less than the weight and wrong. Before using a formula, check which angle the problem gives you.

Checking an answer without the formula

Three quick checks find many mistakes. If both ropes hang straight down (), each tension should be . If the two angles are equal, and should be equal. And as a rope goes flat, its tension should grow without limit; at each tension should be more than . The test puts the general formula through all three.

[12]lines 453–460

Try one

A sign hangs from two wires at and above the horizontal. Which wire carries more tension, and how much more? Work it out before running the cell: the wire's tension has on top, the wire's has , and both have underneath.

[13]lines 471–473

The wire carries more tension: , against for the wire (sign, in that order: first). The steeper wire carries more, as before.

What was checked

The program computes each tension two independent ways, from the formulas and by solving the two balance equations directly, and checks that they agree and that the three forces cancel. The last test checks every number quoted on this page against a third computation outside the program: posts/refs/a-box-hanging-from-two-ropes.py solves the same balance equations with numpy.linalg.solve, and its output is beside it.

[14]lines 489–519