Question

An oscillating block - spring system has a mechanical energy of 1.10 j, and amplitude of 11.0 cm, and a maximum speed of 1.7 m/s. find the spring constant.

Answer 1

The total mechanical energy of the block-spring system is given by the sum of the potential energy and the kinetic energy of the block:
$E=U+K= \frac{1}{2}kx^2 + \frac{1}{2}mv^2$
where
k is the spring constant
x is the elongation/compression of the spring
m is the mass of the block
v is the speed of the block

At the point of maximum displacement of the spring, the velocity of the block is zero: v=0, so the kinetic energy is zero and the mechanical energy is just potential energy of the spring:
$E= \frac{1}{2}kA^2$ (1)
where we used x=A, the amplitude (which is the maximum displacement of the spring).
Since we know 
A = 11.0 cm= 0.11 m
E = 1.10 J
We can re-arrange (1) to find the spring constant:
$k= \frac{2E}{A^2} = \frac{2 \cdot 1.10 J}{(0.11 m)^2}=181.8 N/m$