Question
A block of mass rests on a horizontal surface. The coefficients of static and kinetic friction are and . A horizontal force is gradually increased from zero. At what applied force does the block start moving, and what is its acceleration just after motion begins? Take .
Solution — Step by Step
The block stays at rest until applied force exceeds :
The block starts sliding the instant applied force exceeds . So the threshold is (any infinitesimal excess starts motion).
The instant motion starts, friction drops from (static max) to (kinetic).
Net force just after motion begins (applied force still ):
Final answer: Block starts moving at . Initial acceleration is .
Why This Works
The "" inequality is what gives friction its surprising behaviour: a sudden jump in available net force the moment motion begins. This is why pushing a heavy box is hardest at the start.
Drawing the friction-versus-applied-force graph (linear up to , then a jump down to ) makes this lesson permanent.
Alternative Method
Use energy methods over a small displacement. Work done by applied force minus work by kinetic friction equals KE gained. Same answer, but slower for this simple case.
JEE Advanced loves “block on block” friction problems. Start by drawing free-body diagrams for each block separately and identifying which surface’s friction acts on which block.
Common Mistake
Using in Newton’s second law after motion starts. Once the block moves, only matters. Mixing them gives — a “trap” answer that suggests the block doesn’t move at all.