The motion of a springmass system is modeled by the differen

The motion of a spring-mass system is modeled by the differential equation with y\" + 1/8y\' + y = 0, with y(0) = 2 and y\' (0) = 0. (a) Find the position of the mass at time t. (b) Is the motion undamped, under-damped, critically damped or over-damped?

Solution

Since we are given that y\'\' + (1/8)*y\' + y = 0

The Auxillary equation is r² + (1/8)r + 1 = 0 => 8r² + r + 8 = 0

From the above quadratic equation find the the values of r₁ & r₂ Such that the motion is overdamped and the solution is Y(t) = C₁e^r₁t + C₂e^{r₂t -----------------(I)}

==> Since we are given with the intial conditions Y(0) = 2 & Y\'(0) = 0

Calculatins C_{1 &} C₂ from above intial contitons after putting them in equation (I).

You will get the Position of mass at time t