We are looking at number 8 here Here is the answer the book

ough 10 the indicated mass-and-spring sys- ) = 0) in its equi- ms 8 thr In Proble tem is set in motion from rest (xi (0) = librium position forces i ve as a superp 0 (x1(0) = X2(0) 0) with the given external s Ft) and F20) acting on the masses mi and m2, respec- . Find the resulting motion of the system and describe it a superposition of oscillations at three different frequencies. 8. The mass-and-spring system of Problem 2, with F1(t) = 96 cos St, F2(t) 0

Solution

F = F1+F2

F=96cos 5t + 0 (F2 =0 because no damping)

Using Newton\'s 2nd law F=ma in derivative form F= mx¹¹

mx¹¹=96cos 5t + 0

x¹¹=96cos 5t/m .

x¹¹=a/m.. (1)

Using Analytic solutions, (If you are intrested in knowing how I got the below solutions check any basic differential calculus book for derivation)

x1(t) = e i c t = cos(c t) + i sin(c t)

x2(t) = e i c t = cos(c t) + i sin(c t)

Where, c = a/m

For linear combinations don\'t use \"i\"

The solution combining equation (1) will be

As mentioned in the above image.

And the reasons are already answered too in the picture.