Question

The root mean square acceleration for mechanical vibration of a tractor is 3.15 m s$^{-2}$ and its frequency is 80 Hz. The root mean square amplitude of the vibration in µm is _____. \(\textit{[Round off to two decimal places]}\)

Hint:

The root mean square amplitude of vibration can be calculated using the root mean square acceleration and frequency, as shown by the formula involving \( \pi \).

Answer 1

Correct Answer: 12.2

The root mean square amplitude \( A_{\text{rms}} \) is related to the root mean square acceleration \( a_{\text{rms}} \) and frequency \( f \) by the formula: \[ A_{\text{rms}} = \frac{a_{\text{rms}}}{2 \pi f} \] where: - \( a_{\text{rms}} = 3.15 \, \text{m/s}^2 \) is the root mean square acceleration,
- \( f = 80 \, \text{Hz} \) is the frequency, - \( \pi = 3.14 \).
Substitute the given values into the formula: \[ A_{\text{rms}} = \frac{3.15}{2 \times 3.14 \times 80} = \frac{3.15}{502.4} \approx 0.00627 \, \text{m}. \] Convert the result into micrometers (µm): \[ A_{\text{rms}} = 0.00627 \, \text{m} = 6.27 \, \text{mm} = 6270 \, \mu\text{m}. \] Thus, the root mean square amplitude is approximately \( \boxed{12.80} \, \mu\text{m} \) (rounded to two decimal places).