Recall the equation of motion in plane polar coordinates of a particle of mass m actedupon by a central force Fr(r) isr ̈ − (L^2/m^2 r^3) = (1/m) F r ( r ) ,where L = mr2θ ̇ is the angular momentum. Let u(θ) = 1/r(θ). Show that the orbit equation with respect to u(θ) is,d^2u(θ)/dθ^2+ u(θ) = −mFr(1/u)/L^2u^2 Recall the equation of motion in plane polar coordinates of a particle of mass m actedupon by a central force F,(r) isL²1F,(r),m2r3mwhere Lmr²0 is the angular momentum. Let u(0)1/r(0). Show that the orbitequation with respect to u(0) is,du(0)mF,(1/u)+ u(0) :d02L?u?

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Recall the equation of motion in plane polar coordinates of a particle of mass m acted upon by a central force F,(r) is L2 1 -F,(r), m2r3 m where L mr20 is the angular momentum. Let u(0) = 1/r(0). Show that the orbit %3D equation with respect to u(0) is, du(0) + u(0) : mF,(1/u) L²u? d02

Recall the equation of motion in plane polar coordinates of a particle of mass m acted upon by a central force F,(r) is L2 1 -F,(r), m2r3 m where L mr20 is the angular momentum. Let u(0) = 1/r(0). Show that the orbit %3D equation with respect to u(0) is, du(0) + u(0) : mF,(1/u) L²u? d02

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Question

Recall the equation of motion in plane polar coordinates of a particle of mass m acted

upon by a central force Fr(r) is

r ̈ − (L^2/m^2 r^3) = (1/m) F r ( r ) ,

where L = mr2θ ̇ is the angular momentum. Let u(θ) = 1/r(θ). Show that the orbit equation with respect to u(θ) is,

d^2u(θ)/dθ^2+ u(θ) = −mFr(1/u)/L^2u^2

Recall the equation of motion in plane polar coordinates of a particle of mass m acted
upon by a central force F,(r) is
L²
1
F,(r),
m2r3
m
where L
mr²0 is the angular momentum. Let u(0)
1/r(0). Show that the orbit
equation with respect to u(0) is,
du(0)
mF,(1/u)
+ u(0) :
d02
L?u?

Definition Definition Product of the moment of inertia and angular velocity of the rotating body: (L) = Iω Angular momentum is a vector quantity, and it has both magnitude and direction. The magnitude of angular momentum is represented by the length of the vector, and the direction is the same as the direction of angular velocity.

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