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(A) Given the potential energy $U(x) = U_0(1 - \cos ax)$.For small oscillations,we can approximate $\cos ax \approx 1 - \frac{(ax)^2}{2}$.Thus,$U(x) \

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$2\pi \sqrt{\frac{m}{a^2 U_0}}$

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(A) Given the potential energy $U(x) = U_0(1 - \cos ax)$.For small oscillations,we can approximate $\cos ax \approx 1 - \frac{(ax)^2}{2}$.Thus,$U(x) \approx U_0(1 - (1 - \frac{a^2 x^2}{2})) = \frac{1}{2} U_0 a^2 x^2$.Comparing this with the potential energy of a simple harmonic oscillator $U = \frac{1}{2} k x^2$,we identify the effective spring constant $k = U_0 a^2$.The angular frequency of oscillation is $\omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{U_0 a^2}{m}} = a \sqrt{\frac{U_0}{m}}$.The time period $T$ is given by $T = \frac{2\pi}{\omega_0} = \frac{2\pi}{a} \sqrt{\frac{m}{U_0}} = 2\pi \sqrt{\frac{m}{a^2 U_0}}$.

The potential energy of a particle of mass $m$ situated in a unidimensional potential field varies as $U(x) = U_0(1 - \cos ax)$,where $U_0$ and $a$ are constants. The time period of small oscillations of the particle about the mean position is

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