An object with mass \(=500 \mathrm{~kg}\) is attached to a table with four steel legs of diameter \(=0.015 \mathrm{~m}\) and length \(=0.1 \mathrm{~m}\) (Figure P1.10).

a. Derive the differential equation for vibration of the object in vertical direction.

b. Determine the natural frequency of vibration.

c. What is the maximum possible amplitude of vibration so that the maximum vibratory stress in each table leg is less than \(70 \%\) of the ![] yield point stress? For this maximum amplitude, plot the allowable region of initial displacement and the initial velocity of the object.

d. Develop a MATLAB program to solve the governing differential equation. Compare the solution from your program to that obtained analytically for an allowable initial displacement and the initial velocity determined in part (c).

Figure P1.10