To find the coefficient of friction between two pieces of…

To find the coefficient of friction between two pieces of oak, a wooden cube of mass $m = 2\ \mathrm{kg}$ is placed on an oak board, and the board is slowly tilted while being tapped lightly with a hammer. At an angle of $\alpha = 32^\circ$ the cube starts to slide down uniformly. What is the coefficient of friction?

Explanation of Method

Friction coefficient from the sliding angle.

Step 1: Uniform sliding

At constant velocity the forces along the board balance: $mg\sin\alpha = \mu\,mg\cos\alpha$.

\[\mu = \tan\alpha = \tan 32^\circ \approx 0.62\]

$32^\circ$ $mg$ $N$ $\mu N$ $\mu = \tan 32^\circ \approx 0.62$

Sliding uniformly down, the block has zero net force: friction $\mu mg\cos\alpha$ up the slope balances $mg\sin\alpha$ down it. The mass cancels, leaving $\mu = \tan\alpha$.

Result

\[\boxed{\,\mu = \tan\alpha = \tan 32^\circ \approx 0.62\,}\]

Remark

Check. The original result is correct. The tapping jolts the cube loose, so it moves as soon as gravity can keep it sliding; the angle then measures the sliding friction coefficient. Without tapping, the cube would stick until a larger angle (static friction). The mass is not needed.