Here is the solution:
The differential equation that describes this driven harmonic oscillator system is:
mɑ = F - kx
Where:
m is the mass = 16 kg
ɑ is the acceleration
k is the spring constant = 196 N/m
x is the displacement
F is the driving force = 48 sin wt newtons
Solving this differential equation for a given:
mɑ = 48 sin wt - 196x
ɑ = (48 sin wt - 196x)/16
ɑ = 3 sin wt - 12x
Since a = d2x/dt2 , we can separate variables and integrate both sides twice to get:
∫d2x = ∫(3 sin wt - 12x)dt
∫dx = 3∫ sin wt dt - 12 ∫xdt
x = 3(-cos wt)/w - 6x2 + C1t + C2
Solving for x:
x = (3/w)cos wt + C1t + C2
Where C1 and C2 are constants determined by the initial conditions.
Thus, the expression for the displacement, x, in terms of t and w is:
x = (3/w)cos wt + C1t + C2