Question

a-slug mass is hung onto a spring, whereupon the spring is stretched 6 in from its natural length. The mass is n started in motion from the equilibrium position with an initial velocity of 4 ft/sec in the upward direction. Find the subsequent motion of the mass, if the force due to air resistance is -2t lb.

Answer 1

using hook's law

we have k = m g / L

k = 0.5 X 32.17 / 1.280

k = 6.283 sec

the diff equation is

d²x/dt² + 8 dx/dt + 25 x = 0

then roots are

$\lambda$₁ = 3 i - 4

$\lambda$₂ = -3 i - 4

then

x = e^-4t ( C₁ cos3t + C₂ sin3t )

x ( 0 ) = 0

and

dx(0)/dt = 4

there fore

x = (4/3) e^-4t

x $\rightarrow$ 0 , x $\rightarrow$$\oe$