A particle of mass m is moving in a potential well, for which the potential energy is given by $$U(x)=U01$$−sinbx where $$U0$$ and a are constants. Then for small oscillations

Question

A particle of mass m is moving in a potential well, for which the potential energy is  given by $$U(x)=U01$$−sinbx where $$U0$$ and a are constants. Then for small  oscillations

Infinity Learn · Accepted Answer

$$U(x)=U0(1$$−$$sin$$⁡$$bx)F=$$−$$dUdx=$$−$$aU0sin$$⁡bx For small oscillations, $$sin$$⁡bx≈bx⇒ma'$$=$$−$$a2U0x$$⇒a'$$=$$−$$a2U0mx$$ . Since, a'α−x , oscillations are simple harmonic.Hence, $$T=2$$$$π$$$$mb2U0$$ .Speed is maximum when a’ $$=$$ $$0$$. It happens at x $$=$$ $$0$$.$$dvdt=$$−$$a2U0xdvdxdxdt=$$−$$a2U0xvdv=$$−$$a2U0xdx$$∫$$max0$$ $$vdv=$$−$$a2U0$$∫$$0A$$ xdxSince, vmax is unknown, A cannot be found.

A particle of mass m is moving in a potential well, for which the potential energy is given by $U (x) = U_{0} (1 - \sin b x)$ where $U_{0}$ and a are constants. Then for small oscillations

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A particle of mass m is moving in a potential well, for which the potential energy is given by U(x)=U01−sinbx where U0 and a are constants. Then for small oscillations

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A particle of mass m is moving in a potential well, for which the potential energy is given by $U (x) = U_{0} (1 - \sin b x)$ where $U_{0}$ and a are constants. Then for small oscillations