Question 1.10.1: The response of a mass oscillating on a surface is measured ......
The response of a mass oscillating on a surface is measured to be of the form indicated in Figure 1.45. The initial position is measured to be 30 mm from its zero rest position, and the final position is measured to be 3.5 mm from its zero rest position after four cycles of oscillation in 1 s. Determine the coefficient of friction.
First, the frequency of motion is 4 Hz, or 25.13 rad/s, since four cycles were completed in 1 s. The slope of the line of decreasing peaks is
\frac{-30 + 3.5}{1} = -26.5 mm/s
Therefore, from expression (1.111),
–\frac{2μm\mathtt{g}w_{n}}{\pi k} (1.111)
-26.5 mm/s = \frac{-2μm\mathtt{g}w_{n}}{\pi k} = \frac{-2μ\mathtt{g}}{\pi} \frac{w_{n}}{w²_{n}} = \frac{-2μ\mathtt{g}}{\pi w_{n}}
Solving for μ yields
μ = \frac{\pi (25.13 rad/s) (-26.5 mm/s)}{(-2) (9.81 × 10³ mm/s²)} = 0.107
This small value for μ indicates that the surface is probably very smooth or lubricated.