Question

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

Answer 1

Answer:

75%

$\dfrac{A}{\sqrt{2}}$

Explanation:

x = Displacement of spring = $\dfrac{1}{2}A=\dfrac{A}{2}$

k = Spring constant

Total energy of the spring is

$E=\dfrac{1}{2}kA^2$

Elastic potential energy is given by

$U=\dfrac{1}{2}kx^2\\\Rightarrow U=\dfrac{1}{2}k(\dfrac{A}{2})^2\\\Rightarrow U=\dfrac{1}{2}k\dfrac{A^2}{4}\\\Rightarrow U=\dfrac{1}{4}\times \dfrac{1}{2}kA^2\\\Rightarrow U=\dfrac{1}{4}E$

Total energy is given by

$E=U+K\\\Rightarrow K=E-U\\\Rightarrow K=E-\dfrac{1}{4}E\\\Rightarrow K=\dfrac{3}{4}E\\\Rightarrow \dfrac{K}{E}=0.75$

The percentage will be

$\dfrac{K}{E}=0.75\times 100\%\\\Rightarrow \dfrac{K}{E}=75\%$

The required percentage is 75%

According to the given condition

$\dfrac{E}{2}=\dfrac{1}{2}kx^2\\\Rightarrow \dfrac{\dfrac{1}{2}kA^2}{2}=\dfrac{1}{2}kx^2\\\Rightarrow \dfrac{1}{2}kA^2=kx^2\\\Rightarrow x=\sqrt{\dfrac{1}{2}A^2}\\\Rightarrow x=\dfrac{A}{\sqrt{2}}$

So, the energy is half when displacement is $\dfrac{A}{\sqrt{2}}$