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phys documents (dragged) 9 - Chapter 1 Mechanics 1.1...

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Unformatted text preview: Chapter 1 Mechanics 1.1 Point-kinetics in a fixed coordinate system 1.1.1 Definitions The position r , the velocity v and the acceleration a are defined by: r = (x, y, z ), v = (x, y, z ), a = (¨, y , z ). ˙˙˙ x¨¨ The following holds: s(t) = s0 + |v (t)|dt ; r (t) = r0 + v (t)dt ; v (t) = v0 + a(t)dt When the acceleration is constant this gives: v (t) = v 0 + at and s(t) = s0 + v0 t + 1 at2 . 2 For the unit vectors in a direction ⊥ to the orbit e t and parallel to it e n holds: et = dr ˙ v v e˙t = et = en ; en = |v | ds ρ |e˙t | d2 r det 1 dϕ = 2= ; ρ= ds ds ds |k | For the curvature k and the radius of curvature ρ holds: k= 1.1.2 Polar coordinates Polar coordinates are defined by: x = r cos(θ), y = r sin(θ). So, for the unit coordinate vectors holds: ˙ ˙ e˙r = θeθ , e˙θ = −θer ˙ ˙ ¨ The velocity and the acceleration are derived from: r = re r , v = r er + rθ eθ , a = (¨ − rθ2 )er + (2r θ + rθ )eθ . ˙ r ˙˙ 1.2 Relative motion For the motion of a point D w.r.t. a point Q holds: r D = rQ + ω × vQ ˙ with QD = rD − rQ and ω = θ. ω2 ¨ Further holds: α = θ. means that the quantity is defined in a moving system of coordinates. In a moving system holds: v = vQ + v + ω × r and a = aQ + a + α × r + 2ω × v + ω × (ω × r ) with ω × (ω × r ) = −ω 2 r n 1.3 Point-dynamics in a fixed coordinate system 1.3.1 Force, (angular)momentum and energy Newton’s 2nd law connects the force on an object and the resulting acceleration of the object where the momentum is given by p = mv : F (r , v , t) = d(mv ) dv dm m=const dp = =m +v = ma dt dt dt dt ...
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.

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