Question

g A 4 lb weight stretches a spring 2 ft in equilibrium. It is attached to a damping mechanism with constant c. (a) Find the value of c for which the motion is critically damped. (b) If the motion is critically damped, find the displacement y(t) when the spring is initially displaced 6 inches above equilibrium and given a downward velocity of 3 ft/sec.

Answer 1

Given :

Weight = 4 lb

Stretched of a spring in equilibrium,Δ = 2 ft

a). We know that

F = kΔ

4 = k x 2

k = 2 lb/ft

∴ Stiffness of the spring is, k = 2 lb/ft

Motion is critically damped.

`$\epsilon = 1$`

`$\epsilon=(C)/(C_c) = 1$`

`$C=C_c`

`$C=C_c=√(4k_m)`

`$C=\sqrt{4 * 2 * \left((4)/(32.174)\right)}`

C = 0.99729 lb.s/ft.

b). Displacement

`$y(t)=y_0 \sin (\omega t)$`

`$\dot{y}(t)= y_0 \omega \cos (\omega t)$`

`$[\dot{y}(t)]_(mon)= y_0 \omega $`

Initial displacement = 6 inches

`$y_0 = 0.5 \ ft , \ \ \ \dot y (t) = 3 \ ft/sec$`

`$\Rightarrow 3=0.5 * \omega$`

`$\Rightarrow \omega = 6$`

Therefore, displacement :

`$y(t)=y_0 \sin (\omega t)$`

`$y(t)=0.5 \sin (6 t)$`