Q32 of 40 Page 103

A body of mass m is situated in a potential field U(x) = U0 (1-cos αx) when U0 and α are constants. Find the time period of small oscillations

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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