Friction: Application Problems (4)

Question

A block of mass $m = 5\,\text{kg}$ rests on a horizontal surface with $\mu_s = 0.5$ and $\mu_k = 0.4$. A horizontal force $F$ is applied. (a) What is the maximum $F$ before the block slips? (b) If $F = 30\,\text{N}$ is applied, find the acceleration. Take $g = 10\,\text{m/s}^2$.

Solution — Step by Step

(a) Maximum $F$ before slipping: $25\,\text{N}$. (b) Acceleration when $F = 30\,\text{N}$: $2\,\text{m/s}^2$.

Why This Works

Static friction self-adjusts up to its maximum. As long as the applied force is below $\mu_s N$, friction matches it exactly and the block stays put. Once the applied force exceeds this threshold, the block accelerates and kinetic friction (smaller than the maximum static value) takes over.

The key check is whether $F > f_{s,\max}$. If yes, use $f_k$. If no, friction equals $F$ and acceleration is zero. Many students skip this check and always use $f_k$, even when the block isn’t moving.

Alternative Method

Compute net force directly. If the block doesn’t accelerate, friction balances $F$. Test: is $F$ greater than the static threshold? If yes, the block must accelerate. Then $a = (F - \mu_k mg)/m$. Same logic, slightly faster.

Common Mistake

Using $\mu_s$ in the kinetic phase, or vice versa. The fix: write down both $f_{s,\max}$ and $f_k$ at the start, then decide which one applies based on whether the block moves. JEE Main 2024 had a stack-of-blocks variant testing exactly this.