Pulley Rope Length Constraints

Why Pulley Problems Are Related-Rates Problems

A rope running over a pulley connects the motion of different objects. If the rope is taut and cannot stretch, its total length stays fixed. When one part of the rope lengthens, another part must shorten to compensate. That restriction links the speeds of the loads.

The main task is to count the rope segments whose lengths change, write their total as a constant, and differentiate that equation with respect to time. The signs tell you whether connected parts move in the same direction or opposite directions; the coefficients tell you how their speeds compare.

Set Up the Rope-Length Constraint

Assume the rope is taut, inextensible, and has fixed total length. Assume also that the pulleys do not change position or orientation unless a pulley is attached to a moving load. In a basic model, the lengths around the pulleys are constant, so they can be included in a constant term. Only the straight segments whose lengths change need to be written explicitly.

Choose coordinates before writing the equation. In the examples below, distances are measured from fixed reference points, and downward is the positive direction for vertical motion. Thus a positive derivative means downward motion, while a negative derivative means upward motion. State the convention so each rate has an unambiguous meaning.

A segment contributes its length once for each portion of rope that changes length. For a fixed pulley with one load on each end, there are two changing segments, one for each load. For a movable pulley supported by two rope segments, the moving load changes the length of both supporting segments. Its coordinate therefore appears twice in the rope equation.

Differentiate the Constraint

If \(x\) and \(y\) are position coordinates that depend on time \(t\), then differentiating a constant-length equation relates their instantaneous rates. This is the same procedure used for other related-rates constraints: differentiate with respect to time, and interpret each derivative as a signed velocity.

This result requires that the rope’s total length remain fixed and that the listed lengths account for all changing portions. If a segment is diagonal, its length may first need to be expressed in terms of position coordinates; its derivative then follows from the chain rule. In the vertical pulley arrangements here, the variable segment lengths are directly represented by the load coordinates.

Worked Example: A Fixed Pulley Connects Two Loads

The fixed pulley changes the direction of motion but not the speed magnitude in this idealized arrangement. One segment lengthens at exactly the rate the other shortens.

Worked Example: A Movable Pulley Has Two Supporting Segments

Why the Speed Ratio Appears

The coefficient of a coordinate in the rope equation counts how many variable segments change with that coordinate. In the movable-pulley example, moving the pulley down by a small distance lengthens each supporting segment by that distance. The combined increase is twice the pulley’s displacement. To keep the rope length fixed, the free-end segment must shorten by twice as much.

Thus \(\frac{dx}{dt}=-2\frac{dy}{dt}\): the velocities have opposite signs, and the free end’s speed is twice the movable load’s speed. This is a kinematic rate ratio implied by the rope geometry. It does not require a calculation of forces or energy.

More generally, if one moving load changes the lengths of \(k\) supporting segments by the same amount, its coordinate contributes \(ky\) to the total-length equation. The rate equation then contains \(k\frac{dy}{dt}\). This count is valid only when those \(k\) segments really change one-for-one with the load’s displacement and the rest of the rope path is fixed.

Worked Example: Check the Ratio Using Displacements

A Common Mistake: Counting the Movable Pulley Only Once

A tempting but incorrect equation for the movable-pulley arrangement is \(x+y+C=L\). It counts the free-end segment and the load’s position once, as if moving the load changed only one supporting segment. Differentiating that incorrect equation would give \(\frac{dx}{dt}+\frac{dy}{dt}=0\), predicting equal speed magnitudes.

For example, with \(\frac{dx}{dt}=-1.20\) m/s, that mistake predicts \(\frac{dy}{dt}=1.20\) m/s. But the actual rope has two supporting segments, so the correct equation is \(x+2y+C=L\). The correct rate is \(-1.20+2\frac{dy}{dt}=0\), giving \(\frac{dy}{dt}=0.60\) m/s. Substitution verifies the correct relation: \(-1.20+2(0.60)=0\). The incorrect prediction fails it: \(-1.20+2(1.20)=1.20\ne0\).

A second source of error is ignoring the sign convention. With downward positive, upward motion has a negative rate. Use the signed rate in the differentiated equation first; describe direction in words afterward. Replacing a negative rate with its positive speed too early can reverse the conclusion.

A Reliable Setup

Why the Constraint Matters

The rope equation is more than a convenient formula: it expresses the physical restriction that the rope cannot gain or lose length. Differentiating that restriction turns a relationship among positions into a relationship among instantaneous velocities. The number of times a moving load appears in the length equation determines its coefficient in the rate equation, and therefore its mechanical speed ratio with the other end.

Always build the equation from the actual rope path rather than memorizing a ratio. A different path can have a different number of changing segments, and a segment that does not change length should not be counted. Once the geometry and signs are set correctly, differentiation is direct.