Engineering Statics · Dry friction and impending motion
A pushed block tips before it exhausts static friction
Problem
A uniform rectangular block of weight $W=1000\ \mathrm N$ rests on a rough horizontal floor. Its base width in the push direction is $b=0.60\ \mathrm m$, and its height is $a=0.80\ \mathrm m$. A horizontal force $P$ is applied to the right at the top edge. The coefficient of static friction is $\mu_s=0.40$. Treat the contact pressure as a compressive resultant normal force $N\ge0$ whose line of action may shift within the base. For $P>0$, let $F\ge0$ denote the magnitude of the floor friction acting to the left on the block, and let $x_N>0$ denote a rightward shift of the normal resultant from the base center. 1. Assuming full contact, determine the force $P_{slide}$ at which sliding would become impending. 2. Determine the force $P_{tip}$ at which the normal resultant reaches the right edge of the base. Which event occurs first as $P$ grows quasistatically? 3. At $P=300\ \mathrm N$, find $F$, $N$, and the horizontal shift $x_N$ of the normal resultant from the base center. Verify both the friction inequality and $|x_N|<b/2$. 4. Find the value of $\mu_s$ for which sliding and tipping would begin simultaneously. Explain why a calculation beyond the first threshold cannot keep assuming the same full-contact static model. Take counterclockwise moments as positive. This is a finite-contact block problem, not a ladder-contact problem.
Hint
Solve force balance without setting friction equal to its maximum until the sliding threshold is requested.
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