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= (1)/(2)kx^2 + (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= (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= (2E)/(A^2) = (2 \cdot 1.10 J)/((0.11 m)^2)=181.8 N/m`