A metal plug of length l = 4 cm was inserted into a long…

A metal plug of length $l = 4\ \mathrm{cm}$ was inserted into a long metal pipe of radius $r = 2\ \mathrm{cm}$ while the pipe was hot. After cooling, the pipe presses on the plug with a pressure $p = 40\ \mathrm{kPa}$. What axial force is needed to pull the plug out, if the coefficient of friction is $\mu = 0.3$? Assume no permanent deformation.

Explanation of Method

Force to pull a plug out of a pipe.

Step 1: Total normal force

The pressure acts on the plug's curved side surface: $A = 2\pi r l = 2\pi \cdot 0.02 \cdot 0.04 \approx 5.03 \cdot 10^{-3}\ \mathrm{m^2}$.

\[ N = pA \approx 201\ \mathrm{N}. \]

Step 2: Pulling force

\[ F = \mu N = 0.3 \cdot 201. \]

\[F \approx 60.3\ \mathrm{N}\]

$F$ $\mu N$ $l = 4\ \mathrm{cm}$ $2r = 4\ \mathrm{cm}$ $p = 40\ \mathrm{kPa}$

Longitudinal section. The pipe presses on the whole curved surface of the plug with pressure $p$; the total normal force is $p \cdot 2\pi r l$, and friction on it resists the pull.

Result

\[\boxed{\,F \approx 60.3\ \mathrm{N}\,}\]

Remark

Check. The original result is correct. Adding up the normal force as $pA$ works even though the pressure pushes in different directions around the plug, because friction depends only on the size of the normal force at each point.