Lesson–4 : Laws of Motion

STAGE–4 (Part–IV) : Pulley Systems, String Constraints & Acceleration Logic (IIT Core)

🔹 1. WHY PULLEY PROBLEMS ARE DANGEROUS IN IIT

Pulley problems look simple but hide:

Constraint equations
Equal tension assumptions
Acceleration relationships

IIT Truth:
Pulley problems are solved by logic + constraints, not by memorization.

🔹 2. IDEAL STRING & IDEAL PULLEY (VERY IMPORTANT)

Unless stated otherwise, assume:

String is massless and inextensible
Pulley is smooth and massless

Consequences:

Tension is same throughout the string
Length of string remains constant

🔹 3. ATWOOD MACHINE (BASE MODEL)

Two masses m₁ and m₂ are connected by a light string over a smooth pulley.

If m₁ > m₂:

m₁ moves downward
m₂ moves upward

Acceleration:

a = (m₁ − m₂)g / (m₁ + m₂)

Tension:

T = 2m₁m₂g / (m₁ + m₂)

🔹 4. CONSTRAINT EQUATION (IIT HEART)

The length of the string remains constant.

Hence:

If one block moves down by x, the other moves up by x.

Therefore:

a₁ = a₂

Golden Rule:
Acceleration equality comes from string constraint, not assumption.

🔹 5. MOVABLE PULLEY (ADVANCED IIT LOGIC)

In movable pulley systems:

Acceleration of load ≠ acceleration of free end
Tension acts multiple times on the pulley

If pulley supports load by two string segments:

Effective upward force = 2T

Acceleration relation:

a_load = a_{free end} / 2

🔹 6. STEP-BY-STEP IIT METHOD (PULLEY)

Draw diagram clearly
Mark directions of motion
Apply string constraint (length constant)
Write F = ma for each mass
Solve simultaneously

Never jump directly to formulas.

🔹 7. COMMON IIT TRAPS (VERY IMPORTANT)

Assuming same acceleration without constraint logic
Using different tensions without pulley mass
Ignoring multiple tension forces on movable pulley
Wrong sign convention

🔹 8. CONCEPTUAL IIT QUESTIONS

Q: If both masses are equal in Atwood machine, what happens?

Ans: System remains in equilibrium (a = 0).

Q: If pulley has mass, will tensions be equal?

Ans: No. Rotational dynamics comes into play.

🔹 9. TRANSITION TO NEXT LEVEL

Next level includes:

Pulley + inclined plane
Pulley + friction
Non-ideal pulley (rotation)

✅ STAGE–4 (Part–IV) Completed
Pulley Fear DESTROYED 🚀

MIND GROW Magazine