Question

When the displacement of a mass on a spring is 12A the half of the amplitude, what fraction of the mechanical energy is kinetic energy?At what displacement, as a fraction of A, is the mechanical energy half kinetic and half potential?

Answer 1

As the object displacement is half of the amplitude then the potential energy stored in the spring is given by

$U = \frac{1}{2} kx^2$

here we know that

$x = \frac{A}{2}$

$U = \frac{1}{2}k(A/2)^2$

total mechanical energy is given as

$ME = \frac{1}{2}KA^2$

now KE is given by

$KE = ME - U$

$KE = \frac{1}{2}KA^2 - \frac{1}{2}K(A/2)^2$

$KE = \frac{3}{8}KA^2$

now fraction of KE with respect to ME is given as

$f = \frac{KE}{ME} = \frac{3}{4}$

now if the mechanical energy is divided equally in KE and PE

so now we will have

$KE = \frac{1}{2}KA^2 - \frac{1}{2}Kx^2$

$PE = \frac{1}{2}kx^2$

now we have

$KE = PE$

$\frac{1}{2}KA^2 - \frac{1}{2}Kx^2 = \frac{1}{2}Kx^2$

$A^2 - x^2 = x^2$

$x = \frac{A}{\sqrt2}$