Answer:
x (t) = -0.66 * cos (9.8 * t)
Step-by-step explanation:
We have that Newton's second law for a system is:
Knowing m the united mass and k the spring constant
m * (d ^ 2x) / (dt ^ 2) = -k * x
where x (t) is the displacement from the equilibrium position. The equation can be expressed like this:
(d ^ 2x) / (dt ^ 2) + (k / m) * x = 0
Weight units should be converted to mass units as follows:
m = W / g = 24 lb / (32 ft / s ^ 2) = 3/4 slug
We also need to convert inches to feet, to know the stretch, we know that 1 foot is twelve inches, therefore:
4 in * 1 ft / 12 in = 0.33 ft
With Hooke's law we proceed to calculate the spring constant k:
k = W / s = 24 lb / 0.33 ft = 72 lb / ft
Knowing then that m is equal to 3/4 and that k is equal to 72, we can replace in the initial equation:
(d ^ 2x) / (dt ^ 2) + (72 / (3/4)) * x = 0
(d ^ 2x) / (dt ^ 2) + 96x = 0
We know that the solution of a differential equation of the form a (d ^ 2x) / (dt ^ 2) + (w ^ 2) * x = 0 is equal to:
x (t) = C1 * cos (wt) + C2 * sin (wt)
Let w ^ 2 = 96, then w = 96 ^ (1/2) = 9.8
Replacing
x (t) = C1 * cos (9.8 * t) + C2 * sin (9.8 * t)
We have that the initial conditions are x (0) = - 8 in, which is equal to -8/12 ft = -0.66 ft
x (0) = -0.66 and x '(0) = 0 ft / s
Replacing we have:
x (t) = -0.66 * cos (9.8 * t) + 0 * sin (9.8 * t)
Then the equation would be
x (t) = -0.66 * cos (9.8 * t)
