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capa10solutions

capa10solutions - Solution Derivations for Capa#10 1 The...

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Solution Derivations for Capa #10 1) The flywheel of a steam engine runs with a constant angular speed of 172 rev/min . When steam is shut off, the friction of the bearings and the air brings the wheel to rest in 1 . 0 hours. What is the magnitude of the constant angular acceleration of the wheel? (Answer in rev/min 2 ) ω 0 = Given t = Given in hours, convert to minutes. Since ω is given in rev min , we can directly substitute into the angular kinematics equations. For this problem, ω = ω 0 + αt comes in handy. Solving for α , α = ω - ω 0 t In this case, ω (the final rotational speed) is zero since the engine flywheel stops. So, α = - ω 0 t . Units are rev min 2 and CAPA is looking for the magnitude of the answer. 2) How many rotations does the wheel make before coming to rest? (No units required) For this problem, remember from translational kinematics that x = ¯ vt . Similarly, in rotational kinematics, θ = ¯ ωt . Average angular speed is given by 1 2 ( ω + ω 0 ). Thus, the equation becomes θ = 1 2 ( ω + ω 0 ) t Where ω is zero since the flywheel comes to rest and t is in minutes. Simplifying, θ = 1 2 ω 0 t 3) What is the magnitude of the tangential component of the linear acceleration of a particle that is located at a distance of 37 cm from the axis of rotation when the flywheel is turning at 86 rev/min ? This problem may be a little confusing because it does not say that it is still related to problem 1. In this problem you are asked to find the tangential com- ponent of the acceleration. This is given by a t = But α was found in the first problem in units rev min 2 . The radius is given in this problem in cm. Note that the angular speed given has no meaning in this 1

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