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d=Rsinαd equals the fraction with numerator cap R and denominator sine alpha end-fraction

d2Udx2=2U0[3d2x4−2dx3]the fraction with numerator d squared cap U and denominator d x squared end-fraction equals 2 cap U sub 0 open bracket the fraction with numerator 3 d squared and denominator x to the fourth power end-fraction minus the fraction with numerator 2 d and denominator x cubed end-fraction close bracket Evaluating this expression at

From non-inertial frames to complex rotational dynamics. d=Rsinαd equals the fraction with numerator cap R

Fcentrifugal=Mω2Rsinθ⟹Ucentrifugal=−∫Fcentrifugal⋅d(Rsinθ)=−12Mω2R2sin2θcap F sub c e n t r i f u g a l end-sub equals cap M omega squared cap R sine theta ⟹ cap U sub c e n t r i f u g a l end-sub equals negative integral of cap F sub c e n t r i f u g a l end-sub center dot d open paren cap R sine theta close paren equals negative one-half cap M omega squared cap R squared sine squared theta Total effective potential:

Equilibrium occurs where the conservative force acting on the particle is zero. The force is the negative gradient of the potential energy: : Browse at IPhO Olimpicos Savchenko Solutions p=m(x)v=μxvp

dmdt=λdxdt=λvd m over d t end-fraction equals lambda d x over d t end-fraction equals lambda v Substitute dmdtd m over d t end-fraction into the dynamic force equation:

Since no external forces act on the system, categorized by year.

: The official archive of International Physics Olympiad problems from 1967 to the present, categorized by year. : Browse at IPhO Olimpicos Savchenko Solutions

p=m(x)v=μxvp equals m open paren x close paren v equals mu x v Differentiating momentum with respect to time yields:

d2Ueffdθ2|θ=π=−MgR−Mω2R2=−MR(g+ω2R)the fraction with numerator d squared cap U sub e f f end-sub and denominator d theta squared end-fraction vertical line sub theta equals pi end-sub equals negative cap M g cap R minus cap M omega squared cap R squared equals negative cap M cap R open paren g plus omega squared cap R close paren This value is always negative for any real , so the top position is always . Case 3: At the elevated angle ( ) Substitute into the second derivative formula: