Respuesta :
Answer:
Part(a): The equation of motion is [tex]\bf{x(t) = \dfrac{7}{5}~e^{-2t} - \dfrac{2}{5}~e^{-7t}}[/tex].
Part(b): The equation of motion is [tex]\bf{x(t) = -e^{-2t} + \dfrac{11}{5}~e^{-7t}}[/tex].
Explanation:
If '[tex]m[/tex]' be the mass of the object, '[tex]k[/tex]' be the force constant and '[tex]\beta[/tex]' be the damping constant, then the equation of motion of the particle can be written as
[tex]\dfrac{d^{2}x}{dt^{2}} + \dfrac{\beta}{m} \dfrac{dx}{dt} + \dfrac{k}{m}x= 0.........................................(I)[/tex]
Given [tex]m[/tex] = 1 Kg, [tex]k[/tex] = 14 [tex]N~m^{-1}[/tex], [tex]\beta[/tex] = 9. Substituting these values in equation (I),
[tex]\dfrac{d^{2}x}{dt^{2}} + 9~\dfrac{dx}{dt} + 14~x= 0[/tex]
Taking a trial solution [tex]x(t) = e^{mt}[/tex], the auxiliary equation can be written as
[tex]m^{2} + 9m + 14 = 0............................................................(II)[/tex]
and its solutions are [tex]m_{1} = -2~and~m_{2} = -7[/tex], resulting the general solution
[tex]x(t) = C_{1}~e^{-2t} + C_{2}~e^{-7t}....................................................................(III)[/tex]
The velocity at any instant of time of the mass is
[tex]v(t) = -2C_{1}~e^{-2t} _7~C_{2}~e^{-7t}..............................................................(IV)[/tex]
Part(a):
Given [tex]x(t=0) = 1 m,~and~v(t=0) = 0~m~s^{-1}[/tex]. Substituting these values in equation (III) and (IV),
[tex]&& 1 = C_{1} + C_{2}......................(V)\\&and,& 0 = -2C_{1} - 7C_{2}.......................(VI)[/tex]
Solving equations (V) and (VI), we have
[tex]C_{1} = \dfrac{7}{5}~and~C_{2} = \dfrac{-2}{5}[/tex]
So the equation of motion is
[tex]x(t) = \dfrac{7}{5}~e^{-2t} - \dfrac{2}{5}~e^{-7t}[/tex]
Part(b):
Given [tex]x(t=0) = 1 m,~and~v(t=0) = - 12~m~s^{-1}[/tex]. Substituting these values in equation (III) and (IV),
[tex]&& 1 = C_{1} + C_{2}......................(VII)\\&and,& -12 = -2C_{1} -7C_{2}.......................(VIII)[/tex]
Solving equations (V) and (VI), we have
[tex]C_{1} = -1~and~C_{2} = \dfrac{11}{5}[/tex]
So the equation of motion is
[tex]x(t) = -e^{-2t} + \dfrac{11}{5}~e^{-7t}[/tex]
