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## Homework Statement

A spring and dashpot system is to be designed for a 32lb weight so that the overall system is critically damped.

(a) How must the damping constant γ and spring constant k be related?

(b) Assume the system is to be designed so that the mass, when given an initial velocity of 4 ft/sec from its rest position, will have a maximum displacement of 6 inches. What values of damping constant γ and spring constant k required?

## Homework Equations

## The Attempt at a Solution

For the system to be critically damped, γ^(2) = 4km

(a) γ = sqrt(4km)

k = γ^(2)/4m

(b) the IVP is... my'' + λγ' + ky = 0, y(0) = 6, y'(0) = 4

Since γ^2 = 4km,

The solution of the IVP is... y(t) = c

_{1}e^(λt) + c

_{2}te^(λt)

Imposing the initial conditions, I got c

_{1}= 6

and c

_{2}= 4 + 12 k^(1/2)*m^(-1/2)

Now, how do I solve for the damping constant γ and the spring constant k?

Please help!