Question

A body performing linear SHM has an amplitude of $5\,cm$. Find the ratio of its P.E to K.E when it covers $2\,cm$ from its extreme position.

Answer 1

The simple harmonic motion (S H M) is a special type of periodic motion in which the restoring force on the object which is in motion is directly proportional to the object’s displacement magnitude and its direction is towards the object's equilibrium position.

Formulas used:
The kinetic energy of body performing S H M
$K = \dfrac{1}{2}m{\omega ^2}\left( {{a^2} - {y^2}} \right)$ ………… $\left( 1 \right)$
The potential energy of body performing S H M
$U = \dfrac{1}{2}m{\omega ^2}{y^2}$ ………. $\left( 2 \right)$
Where, $m$ = Mass, $\omega$ = Angular frequency, $a$ = Amplitude and $y$ = Displacement.

Complete step by step answer:
Given: Amplitude $\left( a \right)$ = $5\,cm$.
Displacement $\left( y \right)$ = $2\,cm$.
We know that,
Kinetic energy, $K = \dfrac{1}{2}m{\omega ^2}\left( {{a^2} - {y^2}} \right)$
Substituting the given data in above equation, we get kinetic energy as
$K = \dfrac{1}{2}m{\omega ^2}\left( {{5^2} - {2^2}} \right)$
On simplifying the above equation, we get
$K = \dfrac{1}{2}m{\omega ^2}\left( {25 - 4} \right)$………. $\left( 3 \right)$

We know that, potential energy, $U = \dfrac{1}{2}m{\omega ^2}{y^2}$
Substituting the given data in above equation, we get potential energy as
$U = \dfrac{1}{2}m{\omega ^2}{2^2}$
$\Rightarrow U = \dfrac{1}{2}m{\omega ^2}\left( 4 \right)$ ………… $\left( 4 \right)$
Dividing equation $\left( 4 \right)$ by equation $\left( 3 \right)$ , we get
$\dfrac{U}{K} = \dfrac{{\dfrac{1}{2}m{\omega ^2}\left( 4 \right)}}{{\dfrac{1}{2}m{\omega ^2}\left( {25 - 4} \right)}}$
On simplifying the above equation, we get the ratio of potential energy to that of kinetic energy as,
$\therefore \dfrac{U}{K} = \dfrac{4}{{21}}$

Additional information:
Elastic potential energy: It is the potential energy stored in an object as a result of deformation of an object, for example stretching of a spring.
Restoring force: Restoring force is the force acting in opposition to the force caused by a deformation.
Stable equilibrium point: Stable equilibrium point is the point where the net force on a system is zero, but a small displacement of the mass will cause a restoring force that points towards the equilibrium point.

Note: In simple harmonic motion there are no dissipative forces hence it should be noted that the total energy is the sum of the potential energy and kinetic energy. And also the ratio of potential energy to that of kinetic energy is dimensionless, that is it doesn’t have a unit.