Find its displacement?

Question

Assume that there's no damping, an object stretches a spring 4 inches in equilibrium. Find its displacement for t>0 if it's initially displaced 36 inches above equilibrium and given a downward velocity of 2 ft/s.

 

This is a spring problem in Differential Equations. Please show your work. Thanks. The equation of motion is y"+(k/m)y=0. But how do I find k and m so I can set up the differential equation and solve for y?

Answer 1

Assume that there's no damping, an object stretches a spring 4 inches in equilibrium. Find its displacement for t>0 if it's initially displaced 36 inches above equilibrium and given a downward velocity of 2 ft/s.

This is a spring problem in Differential Equations. Please show your work. Thanks. The equation of motion is y"+(k/m)y=0.

 

When the system is at rest, the sum of the forces =0

By Hooke's Law, Fs = kx                    x = 4 in = 0.333 ft

Therefore, Fs = Fg     kx = mg    k/m = g/x = 32 ft/s²/ 0.333 ft = 96

 

     

y'' = -(k/m)y = -ω²y                

The angular frequency ω = √(k/m) = √96 = 4√6

The solution to this differential equation is 

y = A cos ωt + B sin ωt

At t=0, y = + 36 = A

y = 36 cos ωt + B sin ωt

 

Differentiating,

y' = -ω 36 sin ωt + ωB cos ωt

At t = 0, y' = - 2 ft/s

 -2 =  ωB                B = -2/ω = -2/4√6 = - √6/12 = -0.204

y = 36 cos 4√6t - 0.204 sin 4√6t