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The total mechanical energy of a spring-mass system in simple harmonic motion is $E = \dfrac{1}{2}m{\omega ^2}{A^2}$. Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude $A$ remains the same. The new mechanical energy will:
A) Become $2E$
B) Become $\dfrac{E}{2}$
C) Become $\sqrt 2 E$
D) Remains $E$

Last updated date: 21st Apr 2024
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Hint: A spring-mass system consists of masses connected to springs. The system is oscillating in simple harmonic motion. In this question, we have to find the change in the total energy when the mass is doubled. For that, we have to find the dependence of the total energy on mass. Using that relation we can find the change in total energy.

Formula used:
$T = \dfrac{{2\pi }}{\omega }$(Where$T$is the time period of oscillation, $2\pi $is constant and$\omega $is the angular velocity of the oscillating particle)

Complete step by step solution:
The total mechanical energy of the spring-mass system is given by,
$E = \dfrac{1}{2}m{\omega ^2}{A^2}$
The time period of oscillation for simple harmonic oscillation is given by,
$T = \dfrac{{2\pi }}{\omega }$
From this equation we get
$\dfrac{T}{{2\pi }} = \omega $
For a spring-mass system, the time constant is,
\[T = 2\pi \sqrt {\dfrac{m}{K}} \] (Where $m$is the mass of the particle and $K$is a constant called spring constant)
From this equation we get
$\dfrac{T}{{2\pi }} = \sqrt {\dfrac{m}{K}} $
Comparing equations and
We get, $\dfrac{1}{\omega } = \sqrt {\dfrac{m}{K}} \Rightarrow \omega = \sqrt {\dfrac{K}{m}} $
Substituting the value of $\omega $in
$E = \dfrac{1}{2}m \times \dfrac{K}{m} \times {A^2}$ $\left( {\because {\omega ^2} = \dfrac{K}{m}} \right)$
The equation will become,
$E = \dfrac{1}{2}K{A^2}$
This means that the total energy of the system does not depend on the mass of the particle.
Therefore, the energy will remain the same even when the mass is doubled.

The answer is Option (D): remains $E$.

Note: In the expression for total energy, mass is given but this does not mean that there is a relation between mass and total energy of the system. We have to substitute for the frequency and check whether there is a relation between the mass and the total energy. Choosing the answer by seeing the options without substituting for the value of frequency might go wrong.
The motions, which are repeated at regular intervals of time, are called periodic or harmonic motions. The simplest form of oscillatory motion is called simple harmonic motion. The Spring-mass system is a typical example of simple harmonic motion. The time taken to repeat a periodic motion is called the time period of harmonic motion.