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You draw the forces, write Newton's second law for each block, and still get the wrong answer. In pulley problems the usual culprit is not the forces at all. It is the hidden assumption that every block moves with the same acceleration. This post gives you a method for finding the constraint relation, two worked examples with numbers, and a check that catches errors in seconds.
Where do pulley problems sit in the JEE syllabus?
The word "pulley" does not appear in either JEE syllabus, but the tools do. The JEE (Main) 2026 syllabus lists Newton's second and third laws and equilibrium of concurrent forces in its Laws of Motion unit, and the JEE (Advanced) 2026 brochure lists Newton's laws of motion and frames of reference. NCERT's Class 11 Chapter 4 treats strings as transmitting a tension force. Pulley systems are how both exams test whether you can apply those laws to connected bodies. Check your year's bulletin for the current wording.
Why does a diagram matter so much here?
Rosengrant, Van Heuvelen and Etkina (2009) followed students in a course that stressed free-body diagrams for two years. Students who drew the diagrams correctly were significantly more successful at getting the right answer, and high-achieving students also used the diagram to check their work.
There is a caution. Heckler (2009) gave 891 introductory students simple force problems. Those told to draw a force diagram were less likely to solve them than those left to choose a method, and more likely to draw incorrect forces. The diagram helps when you use it to think, not when it becomes a box to tick before the "real" work.
How do you find a constraint relation?
Use the same four steps every time.
Pick a fixed reference, such as the ceiling or the fixed pulley, and write each moving body's position from it (x_A, y_B and so on).
Write the length of each string as the sum of its straight segments in terms of those positions. Ignore the bits wrapped round pulleys; they are constant.
Set each length constant and differentiate twice. This gives a relation between accelerations.
Keep the signs. If a coordinate increases downwards, a positive acceleration means downwards.
Worked example 1: Atwood machine
Masses of 3 kg and 5 kg hang over a light, frictionless fixed pulley. Take g = 10 m/s².
Constraint: one string, so if the 5 kg block goes down by some distance the 3 kg block goes up by the same distance. Both have acceleration a.
5 kg block: 50 − T = 5a. 3 kg block: T − 30 = 3a.
Adding: 20 = 8a, so a = 2.5 m/s². Then T = 30 + 3 × 2.5 = 37.5 N.
Here the "same acceleration" assumption is correct. The next example is where it breaks.
Worked example 2: block on a table pulling a movable pulley
Block A (2 kg) sits on a smooth table. A light string runs from A over a fixed pulley at the table's edge, down and round a light movable pulley that carries block B (4 kg), and up to a hook in the ceiling. Take g = 10 m/s².
Constraint. Let x be the length of string between A and the edge pulley, and y the depth of the movable pulley below the edge. String length = x + 2y + constant. Differentiating twice: a_x + 2a_y = 0. When A moves towards the edge (x shrinks), B goes down at half A's rate: a_A = 2a_B.
Block A: T = 2a_A = 4a_B.
Block B: two string segments pull up, so 40 − 2T = 4a_B.
Substituting: 40 − 8a_B = 4a_B, so a_B = 10/3 ≈ 3.33 m/s², a_A ≈ 6.67 m/s² and T = 40/3 ≈ 13.3 N.
Check B: 40 − 2 × 13.33 = 13.33 N, and 4 × 3.33 = 13.33 N. It balances.
What mistakes should you watch for?
One acceleration for everything. Only true when a single string runs over fixed pulleys.
Forgetting the second segment. A movable pulley with the string looped under it feels 2T upwards, not T.
Mixed sign conventions. Pick "down is positive" for all hanging blocks, or "in the direction of motion" for each, and stick to it.
Same tension in different strings. Tension is the same along one light string over light, frictionless pulleys. A second string has its own tension.
If your free-body diagrams are the weak link, our guide to drawing the diagram first walks through the habit, and the post on friction and block problems adds friction to the same set-ups.
A practice routine
Solve five pulley problems where you write only the constraint relation, not the full solution. Check each with the power check.
Solve the same five fully. Compare: did any error come from the constraint?
Mix in three problems with two strings, where each string gives its own relation.
When you cannot see why your constraint is wrong, the Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can compare your string-length equation with a full solution.
Next time you open a pulley question, write the string length equation before any force equation.
