A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight.
Determine the value of alpha > zero so x(t) is critically damped.
Determine x(t) if the weight is liberated from it's equilibrium position with a descending speed of 3 feet per second
I know i need to use Hooke's and then Newton's law
The Attempt at a Solution
I'm fully aware of how to solve a second order differential equation.
I also know that, in the end, i have to achieve
x´´(t) + 4ax´(t)+16x(t)=0 ///
Since that second order ODE has no g(x) i just use the homogeneous formula with the initial conditions to create the function i need.
I also know alpha needs to be 2 so this is a critically damped system because, here, the frequency squared is 16 and 2 times the number alongside x´(t) needs to be equal to the frequency (2a=w) => a=2
I found this problem in my DE course at university and i couldn't solve it because i couldn't even get to the problem, i am now trying to find sense to it but i need some directions
Thanks to anyone who made time to read this :D