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Question
A body of mass m is situated in a potential field U(x) = U0 (1 – cos αx) when U0 and α are constants. Find the time period of small oscillations.
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Solution
Given the potential energy associated with the field
U(x) = U0 (1 – cos αx) [∵ For conservative force f, we can write f = `(-du)/(dx)`] ......(i)
Now, Force F = `- (dU(x))/(dx)` .....[We have assumed the field to be conservative]
F = `- d/(dx) (U_0 - U_0 cos ax) = - U_0 a sin ax`
F = `- U_0 a^2x` [∵ For small oscillations ax is small, sin ax ≈ ax] ......(ii)
⇒ F ∝ (– x)
As, U0, a being constant.
∴ Motion is S.H.M for small oscillations.
The standard equation for S.H.M F = `- mω^2x` ......(iii)
Comparing equations (ii) and (iii), we get
`mω^2 = U_0a^2`
`ω^2 = (U_0a^2)/m` or `ω = sqrt((U_0a^2)/m)`
∴ Time period T = `(2pi)/ω = 2pi sqrt(m/(U_0a^2))`
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