Question
A block of mass kg sits on a frictionless table connected by a light inextensible string over a frictionless pulley to a hanging block of mass kg. Find the acceleration of the system and the tension in the string. Take .
This appeared in a JEE Main 2024 PYQ variant and tests whether we can write FBDs cleanly without skipping signs.
Solution — Step by Step
For on the table, the string tension pulls it horizontally toward the pulley. For hanging, gravity pulls down and tension pulls up. Pick the direction of motion as positive — we expect to fall, so down is positive for it and the table block accelerates toward the pulley.
For : . For : . The string is inextensible, so both blocks share the same magnitude .
Adding gives , so
. Quick sanity check: N, which must be true otherwise would not fall.
Final answer: , .
Why This Works
The string constraint is the key idea. Because the string does not stretch, the two blocks have the same speed at every instant, hence the same acceleration magnitude. Newton’s third law means the tension pulling forward is the same magnitude as the tension pulling up — that is what lets us call both tensions .
Once we accept these two ideas, the pulley problem reduces to two scalar equations in two unknowns. The trick of “adding the equations” works because appears with opposite signs in the two FBDs, so it cancels.
Alternative Method
We can treat the system as one composite object. The only external force along the direction of motion is (the table’s normal force and are perpendicular to motion). The total mass being accelerated is . So directly. Use this shortcut only after you can do the FBD method — JEE often gives a twist (friction, incline) where the shortcut breaks.
A common mistake is writing for the hanging block. That assumes the block moves up. Always pick a sign convention first, then stick to it. If comes out negative, the block moves the other way — no harm done.
Common Mistake
Students often forget that for on a smooth horizontal table, gravity and normal force cancel, so the only horizontal force is . Writing here is wrong — gravity does not act along the direction of motion. Always resolve forces along the direction of motion before applying .