Question

A mass m at the end of a spring of spring constant k is undergoing simple harmonic oscillations with amplitude A.

Answer 1

Answer:

a) The potential energy of the system is 1/9 of the total mechanical energy, when $x= \frac{1}{3}\cdot A$.

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 ($E$), measured in joules, is the sum of the translational kinetic energy ($K$), measured in joules, and elastic potential energy ($U$), measured in joules. That is:

$E = K + U$ (1)

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

$K = \frac{1}{2}\cdot m \cdot v^{2}$ (2)

$U = \frac{1}{2}\cdot k\cdot x^{2}$ (3)

Where:

$m$ - Mass, measured in kilograms.

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

$k$ - Spring constant, measured in newtons per meter.

$x$ - Elongation of the spring, measured in meters.

If we know that $U = \frac{1}{9}\cdot E$, $k = k$ and $E = \frac{1}{2}\cdot k \cdot A^{2}$, then:

$\frac{1}{18}\cdot k\cdot A^{2} = \frac{1}{2}\cdot k\cdot x^{2}$

$\frac{1}{9}\cdot A^{2} = x^{2}$

$x= \frac{1}{3}\cdot A$

The potential energy of the system is 1/9 of the total mechanical energy, when $x= \frac{1}{3}\cdot A$.

b) If we know that $k = k$, $x = \frac{1}{2}\cdot A$ and $E = \frac{1}{2}\cdot k \cdot A^{2}$, then the equation of energy conservation associated with the system is:

$\frac{1}{2}\cdot k\cdot A^{2} = \frac{1}{4}\cdot k\cdot A^{2}+K$

$K = \frac{1}{4}\cdot k\cdot A^{2}$

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.