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.

Answer 1

Given the potential energy associated with the field

U(x) = U₀ (1 – cos αx)  [∵ For conservative force f, we can write f = ${\displaystyle \frac{-du}{dx}}$] ......(i)

Now, Force F = ${\displaystyle -\frac{dU\left(x\right)}{dx}}$  .....[We have assumed the field to be conservative]

F = ${\displaystyle -\frac{d}{dx}(U_0-U_0\cos ax)=-U_{0}a\sin ax}$

F = ${\displaystyle -U_0{a}^{2}x}$  [∵ 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 = ${\displaystyle -m{\omega }^{2}x}$  ......(iii)

Comparing equations (ii) and (iii), we get

${\displaystyle m{\omega }^2=U_0{a}^2}$

${\displaystyle {\omega }^2=\frac{U_0{a}^2}{m}}$ or ${\displaystyle \omega =\sqrt{\frac{U_0{a}^2}{m}}}$

∴ Time period T = ${\displaystyle \frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{U_0{a}^2}}}$