Respuesta :
Answer:
t = 025 s
Explanation:
We know
weight, W = 4 pounds
spring constant, k = 2 lb/ft
Positive damping, β = 1
Therefore mass, m = W / g
m = 4 / 32
= 1 / 8 slug
From Newtons 2nd law
[tex]\frac{d^{2}x}{dt^{2}}=-kx-\beta .\frac{dx}{dt}[/tex]
where x(t) is the displacement from the mean or equilibrium position. The equation can be written as
[tex]\frac{d^{2}x}{dt^{2}}+\frac{\beta }{m}.\frac{dx}{dt}+\frac{k}{m}x=0[/tex]
Substituting the values, the DE becomes
[tex]\frac{d^{2}x}{dt^{2}}+8\frac{dx}{dt}+16x=0[/tex]
Now the equation is
[tex]m^{2}+8m+16=0[/tex]
and on solving the roots are
[tex]m_{1}[/tex] = [tex]m_{2}[/tex] = -4
Therefore the general solution is [tex]x(t)=e^{-4t}\left ( c_{1}+c_{2}t \right )[/tex]
Now for initial condition x(0) = -1 ft
x'(0)= 8 ft/s
Now we can find the equation of motion becomes,
[tex]x(t)=e^{-4t}\left ( -1+4t \right )[/tex]
Therefore, the mass passes through the equilibrium when
x(t) = 0
[tex]e^{-4t}\left ( -1+4t \right )[/tex] = 0
-1+4t = 0
t = [tex]\frac{1}{4}[/tex]
= 0.25 s
