A particle of mass m moves in a one-dimensional potential energy U(x) = -ax^2 + bx^4, where 'a' and 'b' are positive constants. The angular

Q: A particle of mass $m$ moves in a one-dimensional potential energy $U(x) = -ax^2 + bx^4,$ where $'a'$ and $'b'$ are positive constants. The angular frequency of small oscillations about the minima of the potential energy is equal to

b $\frac{d V(x)}{d x}=\frac{d}{d x}\left(-a x^{2}+b x^{4}\right)$ $=-2 a x+4 b x^{3}=x\left(4 b x^{2}-2 a\right)=0$ $x_{1}=0$ $x_{2}=\sqrt{a / 2 b}$ $F=-k x$ $\frac{d F}{d x}=k=\frac{d^{2} V(x)}{d x^{2}}$ $k=\frac{d}{d x}\left(-2 a x+4 b x^{3}\right)=-2 a+12 b x^{2}$ $k(x=0)=-2 a$ $k(x=\sqrt{a / 2 b})=-2 a+6 a=4 a$ $\omega=\sqrt{\frac{k}{m}}$ $\omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{4 a}{m}}$ $\omega=2 \sqrt{\frac{a}{m}}$

SECTION - A [PHYSICS - MCQ]

GSEB - English Medium

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

A$\pi \,\sqrt {\frac{a}{{2b}}} $
B$2\,\sqrt {\frac{a}{m}} $
C$\sqrt {\frac{{2a}}{m}} $
D$\sqrt {\frac{a}{{2m}}} $

Advanced

STD 11 Science
NEET
STD 11 - 13. oscillations
Physics

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