formula - page 6 of 9 Information Sheets Cartesian unit...

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page 6 of 9 Information Sheets Cartesian unit vectors may be labelled x, y, z or i, j, k s = = arc length of a circle of radius r swept out by angle θ ( θ in radians) ω = d θ/ d t = v/r = angular velocity or speed; ( v = tangential speed for circular motion) α = d ω/ d t = a t /r = angular acceleration with tangential acceleration a t a r = v 2 /r = ω 2 r = radial (centripetal) acceleration towards the centre v = v + at, ω = ω + αt (constant a, α ) v 2 = v 2 + 2 a ( x - x ) , ω 2 = ω 2 + 2 α ( θ - θ ) (constant a, α ) x = x + v t + 1 2 at 2 , θ = θ + ω t + 1 2 αt 2 (constant a, α ) x = 1 2 ( v + v ) t, θ = 1 2 ( ω + ω ) t (constant a, α ) Centre of mass coordinates: x cm = 1 M n X i =1 m i x i , y cm = 1 M n X i =1 m i y i , z cm = 1 M n X i =1 m i z i Moment of Inertia: I = n X i =1 m i r 2 i = Z r 2 dm , Parallel Axis Theorem: I h = I cm + Mh 2 ~ τ = ~ r × ~ F, τ = rF sin φ, X τ = Kinetic energy = 1 2 I cm ω 2 + 1 2 Mv 2 cm (general), or 1 2 2 for a fixed axis For rolling motion with no slipping, x cm = θR, v cm = ωR, a cm = αR (In the above expressions, ω
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