Question

A spring is attached to a support, it has a suspended mass of 2 Kg and the spring constant (elasticity constant) is 4 N/m.

System is at rest when the support begins to oscillate according to the expression

$h(t)=2cos(3t)$

Determine

a) Differential equation of motion if the whole system is in a medium opposes a resistance force equal to 6 times the instantaneous velocity

b) Equation of motion (consider that the weight is at rest in equilibrium position when the spring starts to oscillate)

c) Graph of equation of motion

Answer 1

(a) differential equation Let at any internet when mass oscillate than force acting on it are restoring force 4 resistive force m dh/dt^2 -kh - cv or d^2h/dt^2 = (-k/m)^n - 6/m dh/dt Rightarrow dh/dt^2 + 3 dh/dt + 2h = 0 (b) Now since b > w_0 H = v/p xp (-bt) sin h pt h = v_0/1/2 exp (-3t/2) sin h(1/2t) = 2 v_0 emp (-3t/2) sin h (t/2) (c) where w^2 = k/m = 4/2 = 2, and b = k/2m = 3/2 dance system is over damped it solution will where Beta = squareroot b^2 - w_0^2 = squareroot (3/2)^2 - 2 = 1/2 v_0 = initial velocity at t = 0