Use the work-energy theorem.
The total work W performed on the block is done by friction and the restoring force in the spring, both of which oppose the block's motion. By Newton's second law,
• the net horizontal force on the block is
∑ F (hor) = -f - s = ma
where f = magnitude of friction, s = mag. of restoring force, m = 1.80 kg, and a is the acceleration of the block; and
• the net vertical force on the block is
∑ F (ver) = n - mg = 0
where n = mag. of normal force.
It follows that
n = mg ⇒ f = 0.56n = 0.56mg
and so the work done by friction as the spring is compressed by 11 cm = 0.110 m is
W (friction) = -0.56mg (0.110 m) = -1.09 J
The work done by the spring as it compresses is
W (restoring) = -1/2 k (0.110 m)²
where k is the spring constant.
Then the total work is
W = -1.09 J - 1/2 k (0.110 m)²
By the work-energy theorem, this work is equal to the change in the block's kinetic energy,
W = ∆K
Since it comes to a stop, we have
∆K = 0 - 1/2 mv² = -1/2 (1.80 kg) (2.00 m/s)² = -3.60 J
So, we have
-1.09 J - 1/2 k (0.110 m)² = -3.60 J
and solving for k gives
k = (-3.60 J + 1.09 J) / (-1/2 (0.110 m)²)
k ≈ 415 N/m
