Ex2. A 4 foot spring measures 8 feet long after an 8 lb. weight is attached to it. The medium through which it moves offers a resistance numerically equal to times the instantaneous velocity. Find the equation of motion if the weight is released from the equilibrium position with a downward velocity of 5 ft/sec. Find the time at which the weight attains its extreme displacement from the equilibrium position.

Solution:

Mass=weight/gravity, so mass m=8/32 or m=1/4. F(s)=ks would be written 8=k*4, so k=2 and the equation is:

The characteristic equation is:whose solution is a double root .

The general solution is.

The IC y(0)=0 è C₁=0.

Note and y’(0)=5 makes C₂=5 so our solution is .

Extreme displacement will occur when y’=0, so solving

we get.

Substituting back into the equation we get extreme displacementfeet.