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A body of mass m is situated in a potent...

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.

Text Solution

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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)))`.
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