Question

A mass weighing 17 lb stretches a spring 7 in. The mass is attached to a viscous damper with damping constant 2 lb *s/ft. The mass is pushed upward, contracting the spring a distance of 2 in, and then set into motion with a downward velocity of 2 in/s. Determine the position u of the mass at any time t. Use 32 ft/s^2 as the acceleration due to gravity. Pay close attention to the units. Leave answer in terms of exact numbers(no decimals).

Answer 1

weight is w=17 pound

gravity is 32 m/s²

mass is given by

slug

.

a force of 17 pounds, stretches a spring 7 inch = 7/12 feet

so x=7/12

from the Hooke's law, spring constant k is

.

.damping constant is

DE is given by

find roots

for complex roots general solution is

....................(1)

here mass is initially released from rest from a point 2 inches above the equilibrium position.

so y(0)=-2

...........put it back in equation 1

.

......................(2)

take derivative

here initial velocity is so y'(0)=2

.................put it back in equation 2

.

.......................answer in inches

or

.......................answer in feet