A 4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to 2 times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 3 ft/s. (Use g

Explanation:

From the given information concerning the spring-mass system:

Let us apply Hooke law.

Then, we have:

mg = ks

8 = k4

k = 8/4

k = 2

Provided that the mass weighing 8 lbs is attached to a spring.

Then, we can divide it by gravity 32 ft/s².

∴

m = 8/32

m = 1/4 slugs

The medium that offers the damping force

Now, let us set up a differential equation that explains the motion of the spring-mass system.

The general equation is:

where;

k = 2,  and

Then;

By solving the above equation, the auxiliary equation is:

Using quadratic formula:

Since this is a repeated root, the solution to their differential equation took the form.

From the initial condition.

At equilibrium position where the mass is being from:

x(0) = 0

Also, at the downward velocity of 3 ft/s

x'(0) = 3

Then, at the first initial condition:

At the second initial condition;

where;

x'(0) = 3

Replacing in the constraints, the equation of the motion is: