Properties of Friction — Question 8

Question 8

Hard uncertainty bounds produce a guaranteed threshold

Take $g=10$ m/s$^2$. Use ideal dry friction, $|f_s|\le\mu_sN$ and $|f_k|=\mu_kN$. A crate mass lies in $[9,11]$ kg and its static coefficient lies in $[0.30,0.40]$. Every combination in these hard bounds is possible. A horizontal pull P acts on a level floor. Treat equality as impending slip.

1. Find the possible range of slip thresholds.

2. Find pulls that guarantee no forced slipping for every allowed crate.

3. Find pulls that guarantee exceeding the static limit; explain what a 35 N pull establishes.

Question 8 - Solution

Hard uncertainty bounds produce a guaranteed threshold

(a) Use monotonicity of the threshold.

The threshold $P_*=\mu_smg$ increases with both positive uncertain inputs. Independent bounds permit both extreme combinations: $$P_{*,\min}=.30(9)(10)=27\ \mathrm N,\qquad P_{*,\max}=.40(11)(10)=44\ \mathrm N.$$ $$\boxed{P_*\in[27,44]\ \mathrm N}.$$

(b) Protect against the smallest capacity.

To support every possible crate, stay below the smallest threshold: $$\boxed{0\leq P\leq27\ \mathrm N}.$$ At 27 N, the weakest allowed crate is at impending slip; strict margin for every crate requires $P<27$ N.

(c) Exceed the largest capacity.

To exceed every possible static limit, use $$\boxed{P>44\ \mathrm N}.$$ For the proposed pull, $27<35<44$ N: some permitted crates must slip while others can remain at rest. Independent hard bounds give no probability for either outcome.