question_answer 57)

                  A body of mass m is situated in a potential field \[U(x)={{U}_{0}}(1-\cos \,\alpha x)\] when \[{{U}_{0}}\] and a are constants. Find the time period of small oscillations.

Answer:

                  \[U(x)=\text{ }{{U}_{0}}(1\text{ }\cos \,\,\alpha x)\]                 \[\therefore \] \[F=-\frac{dU}{dx}\,=-\frac{d}{dx}\,[{{U}_{0}}-{{U}_{0}}\,\cos \,\alpha x]\]                 \[=0-{{U}_{0}}\,\alpha \,\sin \,\alpha x\]                 If \[(\alpha x)\] is small, then \[\sin \,\alpha x=\alpha x\]                 \[\therefore \] \[F=-\,{{U}_{0}}\,{{\alpha }^{2}}x\]                 and acceleration, \[a=\,\frac{F}{m}\,=-\left( \frac{{{U}_{0}}{{\alpha }^{2}}}{m} \right)\,x\]                 Hence, motion is S.H.M.                 \[\therefore \] \[\,T=2\pi \,\sqrt{\frac{m}{{{U}_{0}}{{\alpha }^{2}}}}\]