Question

A mass weighing 16 pounds stretches a spring 8/3 feet. The mass is initially released from rest from a point 5 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 1/2 the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t) = 20 cos(3t). (Use g = 32 ft/s2 for the acceleration due to gravity.)

Answer 1

weight is w=16 pound

gravity is 32 m/s²

mass is given by

slug

.

a force of 16 pounds, stretches a spring 8/3 feet

so x=8/3

from the Hooke's law, spring constant k is

.

.

damping force numerically equal to 1/2 times the instantaneous velocity

so damping constant is

.

force is

DE is given by

find roots

for complex roots complementary solution is

here we have

so assume that a particular solution is

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

put all values in DE

.

.

compare coefficient both sides

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

.

.

general solution is

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

here mass is initially released from rest from a point 5 below above the equilibrium position.

so y(0)=5

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

.

....................(3)

.

take derivative

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

.

.................put it back in equation 3

.

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