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Calculate the time period of the oscilla...

Calculate the time period of the oscillation of a particle of mass m moving in the potential defined as `U(x)={{:((1)/(2) kx^(2)",", x lt 0),(mgx"," , g gt 0):}`

Text Solution

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`U=(1)/(2)kx^(2),xlt0`
Force from potential energy, `F=-(dU)/(dx)=(-d)/(dx)[(1)/(2)kx^(2)]`
`=(-1)/(2)k(d)/(dx)(x^(2))=(-1)/(2)k(2x),F=-kx`
According to newton.s 2nd law, `F=ma`
`ma=-kx`
`a=-(kx)/(m)" "...(1)`
Acceleration of particle in SHM,
`a=-omega^(2)x" "...(2)`
`(1)=(2)implies(-kx)/(m)=-omega^(2)x`
`omega^(2)=(k)/(m)impliesomega=sqrt((k)/(m))`
Time period of the oscillation, `T=(2pi)/(omega)`
`T=2pisqrt((m)/(k))`
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