A body of mass m is situated in a potential field U(x)= U_(0) (1-cos alpha x) when U_(0)" and "alpha are constants. Find the time period of small oscillations.

A body of mass m is situated in a potential field U(x)= U_(0) (1-cos a

1.	A body of mass m is situated in a potential field U(x)= U_(0) (1-cos alpha x) when U_(0)" and "alpha are constants. Find the time period of small oscillations.
Answer» Solution :Potential energy associated with field `U(x) = U_(0)[1-cos alpha x]` DIFFERENTIATE w.r.t. to x, `(dU(x))/(dx)= (d)/(dx)[U_(0)(1-cos alpha x)]` `therefore -F = -U_(0) alpha(-SIN alpha x)` `therefore F = -U_(0) alpha sin alpha x` If `alpha x` is very small, then `sin alpha x approx alpha x` `therefore F= -U_(0) alpha (alpha x)= -U_(0) alpha^(2) x"....."(1)` `therefore F propto (-x)""[therefore U_(0), alpha" are constant "]` For small oscillation motion is SHM, Standard form of FORCE EXERTED on BODY performing SHM `F= -m omega^(2) x""".........."(2)` Comparing equation (1) and (2) `m omega^(2) = U_(0) alpha^(2)` `therefore omega^(2) = (U_0)/(m_0)alpha^(2)` `therefore omega = sqrt((U_0)/(m)alpha^(2))` `therefore (2pi)/(T) = sqrt((U_0)/(m)alpha^(2))` `therefore T =2pi sqrt((m)/(U_(0)alpha^(2)))`.