Question

A body of mass m is situated in a potential field U(x) = Uo(1 – cos α x) when U_o and α are constants. Find the time period of small oscillations.

Answer 1

U(x) = Uo(1 − cos αx)

Differentiating both sides with respect to x

\(\frac{dU(x)}{dx}\)= U_o[0 + α sin αx ] = U_oα sin αx

∴ F = − \(\frac{dU(x)}{dx}\) = −U_oα sin α x 

When oscillations are small, sin θ ≈ θ

or sin αx = αx 

∴ F = −U_oα(αx) = −U_oα²x

∴ F = −(U_oα²)  (i)

We know that F = −kx  (ii)

k = U_oα²  {From (i) & (ii)}

∴ T = \(2\pi\sqrt{\frac{m}{k}}=2\pi\sqrt{\frac{m}{U_o\alpha^2}}\)