A mass m at the end of a spring of spring constant k is undergoing simple harmonic oscillations with amplitude A. Part (a) At what positive value of displacement x in terms of A is the potential energy 1/9 of the total mechanical energy?
Part (b) What fraction of the total mechanical energy is kinetic if the displacement is 1/2 the amplitude?
Part (c) By what factor does the maximum kinetic energy change if the amplitude is increased by a factor of 3?

a) The potential energy of the system is 1/9 of the total mechanical energy, when .

b) The fraction of the total mechanical energy that is kinetic if the displacement is 1/2 the amplitude is 1/2.

c) The maximum kinetic energy is increased by a factor of 9.

Explanation:

a) From Mechanical Physics, we remember that the mechanical energy of mass-spring system (), measured in joules, is the sum of the translational kinetic energy (), measured in joules, and elastic potential energy (), measured in joules. That is:

(1)

By definitions of translational kinetic energy and elastic potential energy, we have the following expressions:

(2)

(3)

Where:

- Mass, measured in kilograms.

- Velocity of the mass, measured in meters per second.

- Spring constant, measured in newtons per meter.

- Elongation of the spring, measured in meters.

If we know that , and , then:

The potential energy of the system is 1/9 of the total mechanical energy, when .

b) If we know that , and , then the equation of energy conservation associated with the system is:

The fraction of the total mechanical energy that is kinetic if the displacement is 1/2 the amplitude is 1/2.

c) From the Energy Conservation equation associated with the system, we know that increasing the amplitude by a factor of 3 represents an increase in the elastic potential energy by a factor of 9. Then, the maximum kinetic energy is increased by a factor of 9.

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