Answer to Question #246607 in Mechanics | Relativity for mavado

Question

Question 2

A spring-mass system, with a spring stiffness of 5,000 N/m, is subjected to a harmonic force of 

magnitude 30 N and frequency 20 Hz. The mass is found to vibrate with an amplitude of 0.2 m.

Assuming that vibration starts from rest (x0 = ẋ0 = 0), determine the mass of the system. [10]

Answer 1

The static deflection

$δ_{st} = \frac{F_0}{k} \\ = \frac{30 \;N}{5000 \;M/m} \\ = 0.006 \;m$

The frequency of the motion

$ω= 2 \pi f \\ = 2 \pi (20) \\ = 125.66 \;rad/s$

The natural frequency of motion

$X = \frac{δ_{st}}{(1-(\frac{ω}{ω_n})^2)} \\ 0.2 = \frac{0.006}{(1 -(\frac{125.66}{ω_n})^2)} \\ 1 -(\frac{125.66}{ω_n})^2 = 0.03 \\ \frac{125.66}{ω_n} = 0.9848 \\ ω_n = 127.58 \;rad/s$

The mass of the system:

$ω_n = \sqrt{\frac{k}{m}} \\ 127.58 = \sqrt{\frac{5000}{m}} \\ m = 0.3071 \;kg$