A particle of mass m moves in a conservative force field along x axis where the potential energy U varies with position x as $U = U_{0}$ (1 – cos ax), $U_{0}$ and a are positive constants. Total energy of the particle is $U_{0}$ and initially it is at x = 0.
Which of the following statements are correct?

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Speed of the particle is maximum at x = 0.

Maximum x – coordinate of the particle is $\frac{π}{2 a}$ .

Maximum kinetic energy of the particle is $U_{0}$ .

Particle will be in equilibrium at x = 0.

answer is A, B, C, D.

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Detailed Solution

$U = U_{0} (1 - \cos a x)$
$U = U_{0}$ (i.e., total energy is potential) when
$\cos a x = 0 \Rightarrow a x = \pm \frac{π}{2} \Rightarrow x = \pm \frac{π}{2 a}$
At these positions KE = 0
Force $F = - \frac{d U}{d x} = - U_{0}$ a sin ax
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$∴$ force is negative (i.e., towards O) at $x = \frac{π}{2}$ , and it is positive (i.e., towards O) at $x = - \frac{π}{2}$
Hence, the particle oscillates between A and B.