Respuesta :
Answer with Step-by-step explanation:
Let m be the mass attached and k be the spring constant and [tex]\beta[/tex] be the positive damping constant.The Newton's second law for the system is given by
[tex]m\frac{d^2x}{dt^2}=-kx-\beta \frac{dx}{dt}[/tex]
Where x(t) is the displacement from the equilibrium position.
The equation can be written as
[tex]m\frac{d^2x}{dt^2}+\beta\frac{dx}{dt}+kx=0[/tex]
[tex]\frac{d^2x}{dt^2}+\frac{\beta}{m}\frac{dx}{dt}+\frac{k}{m}x=0[/tex]
Weight=W=6 lb
s=2 feet
[tex]g=32 ft/s^2[/tex]
[tex]m=\frac{W}{g}=\frac{6}{32}=\frac{3}{16} slug[/tex]
[tex]k=\frac{W}{s}=\frac{6}{2}=3 lb/ft[/tex]
[tex]\frac{d^2x}{dt^2}+\frac{16\beta}{3}\frac{dx}{dt}+16 x=0[/tex]
[tex]2\lambda=\frac{16\beta}{3}\implies \lambda=\frac{8\beta}{3}[/tex]
[tex]\omega^2=\frac{k}{m}=16[/tex]
[tex]\lambda^2-\omega^2=\frac{64\beta^2}{9}-16[/tex]
a.The motion is overdamped when [tex]\lambda^2-\omega^2 >0 [/tex] or [tex]\beta > \frac{3}{2}[/tex]
b.The motion is crirtically damped when [tex]\lambda^2-\omega^2 =0 [/tex] or [tex]\beta = \frac{3}{2}[/tex]
c. The motion is underdamped when [tex]\lambda^2-\omega^2 <0[/tex] or [tex]\beta <\frac{3}{2}[/tex]
By using newton's second law for the system we got that for overdamped motion [tex]\lambda^{2}-\omega^{2} > 0 \text { or } \beta > \frac{3}{2}[/tex]
What is Newton's second law for the system ?
Newton's second law for the system is that acceleration of an object produced by a net force is directly proportional to magnitude of the net force, and inversely proportional to the mass of the object.
Given that
Weight[tex]=W=6\ lb[/tex]
[tex]s=2 \ ft[/tex]
[tex]g = 32[/tex] [tex]ft/s^2[/tex]
We know that
[tex]m \frac{d^{2} x}{d t^{2}}=-k x-\beta \frac{d x}{d t}[/tex]
Here x(t) is the displacement from the equilibrium
Now we can write it as
[tex]\begin{aligned}&m \frac{d^{2} x}{d t^{2}}+\beta \frac{d x}{d t}+k x=0 \\&\frac{d^{2} x}{d t^{2}}+\frac{\beta}{m} \frac{d x}{d t}+\frac{k}{m} x=0\end{aligned}[/tex]
Now we can calculate m and k as
[tex]\begin{aligned}&m=\frac{W}{g}=\frac{6}{32}=\frac{3}{16} \mathrm{slug} \\&k=\frac{W}{s}=\frac{6}{2}=3 \mathrm{lb} / \mathrm{ft}\end{aligned}[/tex]
Now we can write equations as
[tex]\begin{aligned}&\frac{d^{2} x}{d t^{2}}+\frac{16 \beta}{3} \frac{d x}{d t}+16 x=0 \\&2 \lambda=\frac{16 \beta}{3} \Longrightarrow \lambda=\frac{8 \beta}{3} \\&\omega^{2}=\frac{k}{m}=16 \\&\lambda^{2}-\omega^{2}=\frac{64 \beta^{2}}{9}-16\end{aligned}[/tex]
Now for overdamped motion
[tex]\lambda^{2}-\omega^{2} > 0 \text { or } \beta > \frac{3}{2}[/tex]
By using newton's second law for the system we got that for overdamped motion [tex]\lambda^{2}-\omega^{2} > 0 \text { or } \beta > \frac{3}{2}[/tex]
To learn more about overdamped motion visit : https://brainly.com/question/15701473
