Ncert Exercises & Solutions

A body of mass m is situated in a potential field U(x) $= U_{0} (1 - \cos α x)$ where $U_{0}$ and $α$ are constants. Find the time period of small oscillations.

Hint: Use the relation between the force and the potential energy.

Step 1: Find the force equation.

Given potential energy associated with the field,

U(x)

$= U_{0} (1 - \cos α x)$ ...(i)

Now, Force, F

$= - \frac{d U (x)}{d x}$

[

$∵$ for a conservatine force F, we can write

$F = \frac{- dU}{dx}$ ]

[We have assumed the field to be conservative]

$F = - \frac{d}{d x} (U_{0} - U_{0} \cos α x) = - U_{0} α \sin α x$

$F = - U_{0} α^{2} x . . . (i i)$

$\Rightarrow F \propto (- x)$

$U_{0}, α$ are constant,

$∴$ Motion is SHM for small oscillations.

Step 2: Find the time period of oscillations.

Standard equation for SHM,

$F = - m ω^{2} x . . . (i i i)$

Comparing Eqs. (ii) and (iii), we get,

$m ω^{2} = U_{0} α^{2}$

$ω^{2} = \frac{U_{0} α^{2}}{m} o r ω = \sqrt{\frac{U_{0} α^{2}}{m}}$

$∴$ Time period,

$T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{U_{0} α^{2}}}$

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