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A. $b$ and $m$ alone

B. $b$, $a$ and $m$ alone

C. $b$ alone

D. $b$ and $a$ alone

Answer

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$F = - \dfrac{{dU}}{{dx}}$; Representing force(F) is the (negative)derivative of Potential energy($U$).

$a\left( t \right) = - {\omega ^2}x\left( t \right)$, where $a(t)$ is acceleration with respect to time, $\omega$ is angular velocity and $x(t)$ is the displacement function.

Here, a particle is performing a simple harmonic motion in one-dimensional motion which has potential energy at point $x$ from the mean position is $U$.

We have,

Potential energy, \[U = a + b{x^2}\]

Where, $a$ and $b$ are constants

Now Force, $F = - \dfrac{{dU}}{{dx}}$

By putting the value of $U$in the above equation, we get

$

F = - \dfrac{{d \cdot \left( {a + b{x^2}} \right)}}{{dx}} \\

F = - \left[ {\dfrac{d}{{dx}}\left( a \right) + \dfrac{d}{{dx}}\left( {b{x^2}} \right)} \right] \\

$

Since the derivative of constant is zero, so

$\dfrac{d}{{dx}}\left( a \right) = 0$

By substituting this, we have

$ \Rightarrow F = 0 - b \cdot \dfrac{d}{{dx}}\left( {{x^2}} \right)$

$ \Rightarrow F = 0 - b \cdot 2x$ (derivative of ${x^n}$ is $n{x^{\left( {n - 1} \right)}}$)

$ \Rightarrow F = - 2bx \cdot \cdot \cdot \cdot \cdot \cdot \left( 1 \right)$

We know that, $F = ma$ from second law of newton where,

$m$=mass of particle

$a$=acceleration of particle

So,

$ \Rightarrow a = \dfrac{F}{m}$

Put the value of $F$that we get in equation $\left( 1 \right)$

$ \Rightarrow a = \dfrac{F}{m} = - \dfrac{{2b}}{m}x \cdot \cdot \cdot \cdot \cdot \cdot \left( 2 \right)$

The acceleration of a particle performing simple harmonic motion is given by,

$a\left( t \right) = - {\omega ^2}x\left( t \right) \cdot \cdot \cdot \cdot \cdot \cdot \left( 3 \right)$

Here, $\omega $ is the angular velocity of a particle.

By comparing $\left( 2 \right)$and$\left( 3 \right)$, we get

$

\Rightarrow {\omega ^2} = \dfrac{{2b}}{m} \\

\Rightarrow \omega = \sqrt {\dfrac{{2b}}{m}} \\

$

So the correct option is A.

One interesting characteristic of the SHM of an object attached to a spring is the angular frequency, and therefore the period and frequency of the motion, depending on only the mass and the force constant, and not on other factors such as the amplitude of the motion.