Question

The physical quantity conserved in simple harmonic motion is:
a. Time period
b. Total energy
c. Potential energy
d. Kinetic energy

Answer 1

Consider the term simple harmonic motion. The mathematical assortment of motion is known as simple harmonic motion. When it is subjected into linear elastic restoring force it is embodied by the oscillation of a mass on spring and it is given by Hooke's law.

Complete step by step answer:
Simple harmonic motion is a unique periodic motion in which the restoring force acts on an object that is in motion. The simple harmonic motion is directly proportional and it is opposite to the displacement vector of the object. This motion represents the resonant frequency and this motion is sinusoidal in time. Simple harmonic motion includes the motion of a simple pendulum. We will always consider the simple pendulum motion as a precise model because the net force that acts on completion of motion of the pendulum and it is proportional to the displacement of the object.

Now let us try to answer the given question. In the simple harmonic motion, there is a continuous exchange of the kinetic and potential. At the equilibrium point the kinetic energy is maximum and the potential energy is zero. At the maximum displacement from the equilibrium point, the kinetic energy is zero and the potential energy is maximum.

From the explanations we can understand that the potential and kinetic energy changes with respect to time. We can calculate the total energy of the simple harmonic motion. The total energy is calculated by adding the kinetic and potential energy. That is,
$\Rightarrow T = K.E + P.E$
The value of the kinetic energy is given as,
$K.E = \dfrac{1}{2}m{\omega ^2}({A^2} - {x^2})$
The value of potential energy is given as,
$P.E = \dfrac{1}{2}m{\omega ^2}{x^2}$
We know the total energy formula. We can substitute the value for the energies. We get,
$T = \dfrac{1}{2}m{\omega ^2}({A^2} - {x^2}) + \dfrac{1}{2}m{\omega ^2}{x^2}$
In the equation we have the $m$, $\omega$ and $x$ as the common terms. But the term $x$has different signs. That is in the first part of the equation the $x$ term has a minus sign. In the second part of the equation the $x$ has a positive sign.
Remember that in addition when we add the same term with different signs, the terms get cancelled.
$\Rightarrow T = \dfrac{1}{2}m{\omega ^2}{A^2} + \dfrac{1}{2}m{\omega ^2}$
Therefore the $x$ term gets cancelled.
We can add the terms in the given equation.
$\Rightarrow T = \dfrac{2}{4}m{\omega ^2}{A^2}$
We can divide the term $\dfrac{2}{4}$to get the answer,
$\Rightarrow T = \dfrac{1}{2}m{\omega ^2}{A^2}$
$\therefore T = \dfrac{1}{2}m{\omega ^2}{A^2}$
The total energy is constant and it is independent of the instantaneous displacement. In other words, the total energy in the SHM is conserved.

Hence, the correct answer is option (B).

Note: Simple harmonic motion is used to represent the model of molecular displacement vibration. It also provides the basis for the convoluted periodic motion with the help of Fourier analysis.