Question

A spring mass oscillator has a total energy ${E_o}$ and an amplitude ${x_o}$. How large will K and U be for it when $x = \dfrac{1}{2}{x_o}$ ?

Answer 1

Hint:A spring mass oscillator follows a simple harmonic oscillator. A spring mass oscillator stores potential energy in the spring as the spring deforms and the mass attached to the spring has kinetic energy with it. Here, x is the distance travelled and ${x_o}$is the original distance.

Complete step by step solution:
The total energy of a spring mass oscillator is given as:
${E_o} = \dfrac{1}{2}kx_o^2$ ;
Step 1: Find the potential energy U:
The potential energy U is given as:
$U = \dfrac{1}{2}k{x^2}$;
Put the value$x = \dfrac{1}{2}{x_o}$(given) in the above equation and solve:
$U = \dfrac{1}{2}k{\left( {\dfrac{1}{2}{x_o}} \right)^2}$;
$ \Rightarrow U = \dfrac{1}{2}k\left( {\dfrac{1}{2}{x_o} \times \dfrac{1}{2}{x_o}} \right)$;
Simplify the above equation:
$ \Rightarrow U = \dfrac{1}{4}\left( {\dfrac{1}{2}k{x_o} \times {x_o}} \right)$;
Here ${E_o} = \dfrac{1}{2}kx_o^2$; Put this relation in the above equation:
$ \Rightarrow U = \dfrac{1}{4}\left( {{E_o}} \right)$;

Step 2: Find K;
The total energy of a system is defined as the summation of Kinetic energy and Potential energy:
${E_o} = K + U$; …(K = Kinetic Energy; U = Potential Energy)
\[ \Rightarrow K = {E_o} - U\];
Put in the given value $U = \dfrac{1}{4}\left( {{E_o}} \right)$ in the above equation:
\[ \Rightarrow K = {E_o} - \dfrac{1}{4}{E_o}\];
\[ \Rightarrow K = \dfrac{3}{4}{E_o}\];

U and K will be large by $U = \dfrac{1}{4}\left( {{E_o}} \right)$and \[K = \dfrac{3}{4}{E_o}\]when $x = \dfrac{1}{2}{x_o}$.

Note:Here for a spring mass oscillator the total energy stored in a spring is in the form of potential energy. Here we have asked about the values of U and K. The total energy of the system would be the addition of potential energy and kinetic energy, we have to apply the conservation of energy as there is no net external force acting on the system. So, the total initial energy of the spring will be equal to the total final energy of the system.