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Unformatted text preview: 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 . hours. What is the magnitude of the constant angular acceleration of the wheel? (Answer in rev/min 2 ) = Given t = Given in hours, convert to minutes. Since is given in rev min , we can directly substitute into the angular kinematics equations. For this problem, = + t comes in handy. Solving for , = - t In this case, (the final rotational speed) is zero since the engine flywheel stops. So, =- 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, = t . Average angular speed is given by 1 2 ( + ). Thus, the equation becomes = 1 2 ( + ) t Where is zero since the flywheel comes to rest and t is in minutes. Simplifying, = 1 2 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 = r But 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 problem. First convert the radius to meters. Then convert to radians. Finally convert the minutes to seconds. (This last step may be omitted because CAPA knows all units. However, I have not tried this). This can all be done with the following step a t = r * 1 m 100 cm * * 2 rad rev * 1 min 60 sec 2 or simply a t = r 100 2 3600 = r 180 000 Remember the sign on ; units, of course, will be in m/s 2 . All of the conversions are built into the formula. Thus, you would enter r in cm and in rev/min 2 (your answer from (1) ) 4) What is the magnitude of the net linear acceleration of the particle in the above question? In this problem, you are asked to find the net linear acceleration which means you need to find the other component of the acceleration. Acceleration is a vector and is composed of both tangential and radial components. The radial component of acceleration is a r = v 2 r But v = r , so a r = w 2 r 2 r = 2 r In question 3, is given in revolutions per minute, so you must convert to radians per second. This can be done by the following: a r = 2 * 2 rad rev !...
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