JEE Physics Constrained Motion Notes Set 02

What is a Constraint Relation?

When bodies are joined by strings, rods or pulleys, or are forced to stay in contact, their positions, velocities and accelerations are related by a geometric constraint. Differentiating the constraint once gives the velocity relation, and again gives the acceleration relation.

Steps to Find the Relation

1. Choose a fixed origin and measure each position from it. 2. Write the condition that the length of the string or rod is constant (or that two points stay in contact). 3. Differentiate with respect to time. 4. Take care with signs: a coordinate that increases when the length of the string beyond the pulley decreases carries a negative sign.

String Constraint

For an inextensible string, the components of the velocities of its ends along the string are equal, and for a pulley system the total rate of change of the lengths of the string segments is zero: \( \sum \frac{dl_i}{dt}=0 \).

Rod Constraint

For a rigid rod, the components of the velocities of its two ends along the rod are equal. For a ladder AB with end A on a wall and end B on the floor making angle \( \alpha \) with the horizontal, \( v_B=v_A\tan\alpha \).

Contact Constraint

Two bodies in contact without separating have equal velocity components along the common normal.

Worked Example

A block on a table is pulled by a string passing over a pulley at height h, the free end being drawn in at speed u. Find the speed of the block when the string makes angle \( \theta \) with the horizontal.
Solution: the length of the string between block and pulley is \( l=\sqrt{x^2+h^2} \), so \( \frac{dl}{dt}=\frac{x}{l}\frac{dx}{dt}=\cos\theta\,v \) and \( \frac{dl}{dt}=u \). Hence \( v=u/\cos\theta \).