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Calculate the first order energy correction for the one dimentional non-degenerate an harmonic oscillator whose harmiltonian id written as; H^ =−(h^2 /(2m)) (d^2 /dx^2 ) +(1/2)kx^2 +(1/5)Υx^3 +(1/(12))βx^4

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{first}\:\mathrm{order}\:\mathrm{energy}\:\mathrm{correction}\: \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{one}\:\mathrm{dimentional}\:\mathrm{non}-\mathrm{degenerate} \\ $$$$\mathrm{an}\:\mathrm{harmonic}\:\mathrm{oscillator}\:\mathrm{whose}\:\mathrm{harmiltonian} \\ $$$$\mathrm{id}\:\mathrm{written}\:\mathrm{as}; \\ $$$$\hat {\mathrm{H}}=−\frac{\mathrm{h}^{\mathrm{2}} }{\mathrm{2}{m}}\:\frac{\mathrm{d}^{\mathrm{2}} }{\mathrm{dx}^{\mathrm{2}} }\:+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{kx}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{5}}\Upsilon\mathrm{x}^{\mathrm{3}} \:+\frac{\mathrm{1}}{\mathrm{12}}\beta\mathrm{x}^{\mathrm{4}} \\ $$

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A particle of mass m moves under the central repulsive force ((mb)/r^3 ) and is initially moving at a distance ′a′ from the origin of a force with velocity ′v′ at right angle to ′a′. show that rcos pθ=a where p =(b/(a^2 v^2 ))+1.

$${A}\:{particle}\:{of}\:{mass}\:{m}\:{moves}\:{under}\:{the}\:{central} \\ $$$${repulsive}\:{force}\:\frac{{mb}}{{r}^{\mathrm{3}} }\:\:{and}\:{is}\:{initially}\:{moving} \\ $$$${at}\:{a}\:{distance}\:'{a}'\:\:{from}\:{the}\:{origin}\:{of}\:\:{a}\:{force} \\ $$$${with}\:{velocity}\:\:'{v}'\:{at}\:{right}\:{angle}\:{to}\:\:'{a}'. \\ $$$${show}\:{that}\:\:\: \\ $$$$\:\:\:\:\:{r}\mathrm{cos}\:{p}\theta={a}\:\:{where}\:{p}\:=\frac{{b}}{{a}^{\mathrm{2}} {v}^{\mathrm{2}} }+\mathrm{1}. \\ $$$$ \\ $$

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A massless spring with a spring constant K=10Nm^(−1) is suspended from rigid support carries a mass m=100g at it lower end the system is subjected to resistive force −pv.it observed that the system oscillates and its energy to one third of it initial value in one and half minutes calculates the value of p

$$\mathrm{A}\:\mathrm{massless}\:\mathrm{spring}\:\mathrm{with}\:\mathrm{a}\:\mathrm{spring}\:\mathrm{constant} \\ $$$$\mathrm{K}=\mathrm{10Nm}^{−\mathrm{1}} \:\mathrm{is}\:\mathrm{suspended}\:\mathrm{from}\:\mathrm{rigid}\:\mathrm{support} \\ $$$$\mathrm{carries}\:\mathrm{a}\:\mathrm{mass}\:\mathrm{m}=\mathrm{100g}\:\mathrm{at}\:\mathrm{it}\:\mathrm{lower}\:\mathrm{end} \\ $$$$\mathrm{the}\:\mathrm{system}\:\mathrm{is}\:\mathrm{subjected}\:\mathrm{to}\:\mathrm{resistive}\:\mathrm{force} \\ $$$$−\mathrm{pv}.\mathrm{it}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{the}\:\mathrm{system}\:\mathrm{oscillates} \\ $$$$\mathrm{and}\:\mathrm{its}\:\mathrm{energy}\:\mathrm{to}\:\mathrm{one}\:\mathrm{third}\:\mathrm{of}\:\mathrm{it}\:\mathrm{initial}\:\mathrm{value} \\ $$$$\mathrm{in}\:\mathrm{one}\:\mathrm{and}\:\mathrm{half}\:\mathrm{minutes}\:\mathrm{calculates}\:\mathrm{the}\: \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\mathrm{p} \\ $$

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