Properties of Friction — Question 10

Question 10

Pulling the upper block changes the limiting subsystem

Take $g=10$ m/s$^2$. Use ideal dry friction, $|f_s|\le\mu_sN$ and $|f_k|=\mu_kN$. A 2 kg upper block rests on a 3 kg lower block on a frictionless floor. Between the blocks, $\mu_s=0.40$ and $\mu_k=0.30$. A horizontal pull P acts rightward on the upper block. Initially both are at rest.

1. Find the largest P permitting common acceleration.

2. For $P=20$ N, find the acceleration of each block after slipping starts.

3. Find the relative displacement after 1 s under this pull, assuming sufficient overlap remains.

Question 10 - Solution

Pulling the upper block changes the limiting subsystem

(a) Use the lower block to find the demand.

The upper contact carries $N=m_ug=20$ N, so $f_{s,\max}=.40(20)=8$ N. Common acceleration would be $a=P/5$. The lower block then demands $$f=m_la=\frac{3P}{5}\leq8\Longrightarrow\boxed{P_{\max}=\frac{40}{3}=13.333\ \mathrm N}.$$

(b) Replace the failed static constraint.

Since $20>P_{\max}$, the upper block slips right relative to the lower. Kinetic friction is $f_k=.30(20)=6$ N: $$\boxed{a_u=\frac{20-6}{2}=7\ \mathrm{m/s^2}},\qquad \boxed{a_l=\frac63=2\ \mathrm{m/s^2}}.$$ The equal and opposite friction forces act on different blocks.

(c) Integrate the relative acceleration.

With zero initial relative velocity, $$a_{u/l}=a_u-a_l=5\ \mathrm{m/s^2},\qquad \boxed{\Delta x_{u/l}=\tfrac12a_{u/l}t^2=\tfrac12(5)(1)^2=2.5\ \mathrm m}.$$ This is the change in relative position while overlap remains sufficient. Reaching an edge changes the contact model.