Compression question Please Help?

Approach:

When the spring is compressed a distance x, it is storing potential energy PEs given by the equation (see Source 1):

(1) PEs = ½ * k * x²

The source of this energy is the work that gravity does Wg on the elevator in pulling it down the distance of free-fall y plus the distance of compression x.

(Note that this is the same as the gravitational PE lost by the elevator in falling the distance y + x.)

Equations and calculations:

(2) Wg = m * g * (y + x)

Equating (1) to (2):

(3) ½ * k * x² = m * g * (y + x)

Rearranging:

(4) (½ * k) * x² - (m * g) * x - (m * g * y) = 0

Substituting values for k, m, g, and y we get:

(5) 3.00e4 * x² - (950 * 9.81) * x - (950 * 9.81 * 20.5) = 0 ==>

(6) 3.00e4 * x² - 9.32e3 * x - 1.91e5 = 0

Solving for the roots of this equation (see Source 2), we get:

x1 = 2.68

x2 = -2.37

So, the amount of compression of the spring = 2.68m <<<=== Answer

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