A particle of mass m in a unidirectional pote | vedclass

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(C) The potential energy of the particle is given by $U(x) = \alpha + 2 \beta x^2$.The force $F$ acting on the particle is the negative gradient of th

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$\pi \sqrt{\frac{m}{\beta}}$

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(C) The potential energy of the particle is given by $U(x) = \alpha + 2 \beta x^2$.The force $F$ acting on the particle is the negative gradient of the potential energy:$F = -\frac{dU(x)}{dx} = -\frac{d}{dx}(\alpha + 2 \beta x^2) = -4 \beta x$.Comparing this with the standard form of the restoring force for simple harmonic motion,$F = -kx$,we identify the effective spring constant as $k = 4 \beta$.The time period $T$ of oscillation for a particle of mass $m$ is given by:$T = 2 \pi \sqrt{\frac{m}{k}}$.Substituting $k = 4 \beta$ into the formula:$T = 2 \pi \sqrt{\frac{m}{4 \beta}} = 2 \pi \cdot \frac{1}{2} \sqrt{\frac{m}{\beta}} = \pi \sqrt{\frac{m}{\beta}}$.Thus,the correct option is $C$.

$A$ particle of mass $m$ in a unidirectional potential field has potential energy $U(x) = \alpha + 2 \beta x^2$,where $\alpha$ and $\beta$ are positive constants. Find its time period of oscillation.

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