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Karnataka Board PUCPUC Science Class 11

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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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.

Short/Brief Note
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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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Chapter 14: Oscillations - Exercises [Page 103]

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NCERT Exemplar Physics [English] Class 11
Chapter 14 Oscillations
Exercises | Q 14.32 | Page 103

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