A particle of mass m moves in one dimension, whose potential energy varies as $$Ux=$$−$$ax2+bx4$$ , where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to

Question

A particle of mass m moves in one dimension, whose potential energy varies as  $$Ux=$$−$$ax2+bx4$$ , where a and  b are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to

Infinity Learn · Accepted Answer

Potential $$energyUx=$$−$$ax2+bx4$$ Force  $$F=$$−$$dUdx=2ax$$−$$4bx3At$$ mean position  $$F=0$$⇒$$2ax=4bx3$$  ∴$$x=a2b$$ $$d2Udx2=$$−$$2a+12bx2=-2a+6a=4a$$   i.e $$+ve$$⇒U is minima at  $$x=a2b$$∴  Force constant $$Keff=Fxor$$ dFdx⇒$$Keff=d2udx2=2a$$−$$12bx2$$ ⇒$$K=2a$$−$$12b$$×$$a2b=4a$$ Now,$$F=ma$$'$$=-$$$$kx$$⇒a'$$=-4am$$.xThis represents equation of SHM.⇒ω$$=4am=2am$$

A particle of mass m moves in one dimension, whose potential energy varies as $U (x) = - a x^{2} + b x^{4}$ , where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to

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A particle of mass m moves in one dimension, whose potential energy varies as Ux=−ax2+bx4 , where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to

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A particle of mass m moves in one dimension, whose potential energy varies as $U (x) = - a x^{2} + b x^{4}$ , where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to