Amass is oscillating with amplitude a at the end of a spring.

part a
how far (in terms of a) is this mass from the equilibrium position of the spring when the elastic potential energy equals one third the kinetic energy
part b
how far is the mass from the equilibrium position when the kinetic energy is 110 of the total energy?

A)

The total energy of the system is equal to the maximum elastic potential energy, that is achieved when the displacement is equal to the amplitude (x=A):

(1)

where k is the spring constant.

The total energy, which is conserved, at any other point of the motion is the sum of elastic potential energy and kinetic energy:

(2)

where x is the displacement, m the mass, and v the speed.

We want to know the displacement x at which the elastic potential energy is 1/3 of the kinetic energy:

Using (2) we can rewrite this as

And using (1), we find

Substituting into the last equation, we find the value of x:

B)

In this case, the kinetic energy is 1/10 of the total energy:

Since we have

we can write

And so we find: