Q. 32

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

Answer :

We know that the derivative of the potential energy gives the force, i.e.,

-----(1)

We are given that,

∴

Putting the above in equation (1) we get

For very small we get,

----(2)

And for simple harmonic motion we know that, -----(3)

Where k is spring constant

Comparing equation 2 and 3

For simple harmonic motion time period is given as follows,

, where k is the spring constant and m is the mass of the system and T is the time period

∴ putting k in the time period equation, we get,

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