Question

A mass weighing 10 pounds stretches a spring 3 inches. This mass is removed and replaces with a mass weighing 20 pounds, which is initially released from a point 4 inches above the equilibrium position with an upward velocity of 54 ft/s. Find the equation of motion, x(t) .

Answer 1

the first mass stretches the spring 3 inches. This will be the equilibrium. therefore both the forces must be equal in magnitude

g = 32 ft/s2

1 ft = 12 inches,

x = 3 inches

m = 10 lb

Now, for the second case,

m = 20 lb, and the spring is displaced through 4 inches, this will be its amplitude of the oscillation.

equation of motion is,

according to Newton's law,

F = ma

the general solution to this equation is

where

at t = 0, x =4 inches

velocity is a derivative of the position.

  

at t = 0 ,

this is the final solution of the equation of motion.