Analytical Mech Homework Solutions 117

# Analytical Mech Homework Solutions 117 - 15 The ball begins...

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Unformatted text preview: 15 The ball begins pure rolling when v = aω … 5 µ g cos θ v = v + xt = v − g ( sin θ + µ cos θ ) t = a t 2 a 2v t= g ( 2sin θ + 7 µ cos θ ) At that time: 2 2v 2 g 4v ( sin θ + µ cos θ ) − x= g ( 2sin θ + 7 µ cosθ ) 2 g 2 ( 2sin θ + 7 µ cos θ )2 x= 8.20 2v 2 ( sin θ + 6 µ cos θ ) g ( 2sin θ + 7 µ cos θ )2 mx = µ mg x = µg x = µ gt , and x = 1 µ gt 2 2 22 ma ω = − µ mga 5 5 µg ω =− 2a 5 µg t ω =ω − 2a Slipping ceases to occur when v = aω … 5 µ gt = aω − µ gt 2 2 aω t= 7g Iω = 2 aω 1 x = µg 2 7 µg 2 a 2ω 2 x= 49 µ g 2 1 1 Let the moments of inertia of A and B be I a = M a a 2 and I b = M bb 2 . 2 2 The angular velocity of A is α while that of B is β − α + φ (remember that in two dimensions, angular velocity is the rate of change of an angle between an line or direction fixed to the body and one fixed in space). For rolling contact, lengths traveled along the perimeters of the disks A and B must be equal to the arc length traveled along the track C. aφ = b β = ( a + 2b ) (α − φ ) 8.21 15 ...
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