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Home » 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.
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.
CBSE | Class 11 | NCERT Exemplar | Oscillations | Physics
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.

Answer:

  F=-\frac{d}{dx}\left[ {{U}_{0}}(1-\cos \alpha x) \right]

F=-{{U}_{0}}\alpha \sin \alpha x

As it is given that  is small, so sin 

\therefore F=-{{U}_{0}}{{\alpha }^{2}}x

and negative sign shows that
For SHM, we know that
Comparing equations, we get,

k={{U}_{0}}{{\alpha }^{2}}

We can write the time period of oscillation as follows:
T=2\pi \sqrt{\frac{m}{k}}
 T=2\pi \sqrt{\frac{m}{{{U}_{0}}{{\alpha }^{2}}}}

 

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