Saturday, 9 February 2019

homework and exercises - Physics, double Atwood machine


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According to this tutorial over here: http://commons.wikimedia.org/wiki/User:Phys1csf4n


It is stated that aA+aB+2aC=0


But I don't understand the logic used. Is there any way to prove this mathematically, instead of logically?


It holds true for the two cases presented, but how do I know that it holds true for every single case? I can not think of the solution for some reason. I do know that T4=T3=T1+T2 and T1=T2, but there is no way to relate the acceleration because the masses are different.



Answer



Consider the position of the axle of the lower pulley. Call its acceleration ap. You need to convince yourself that ap=12(aA+aB) One way to do that is to use linear algebra. First, if aA=aB, the whole system of the lower pulley and two masses is accelerating together, so ap=AA=AB=12(aA+aB). Second, if aA=aB, the rope is just going around the pulley and the pulley is not accelerating at all, so ap=0=12(aA+aB) Now for general aA,aB, resolve them into aA+aB,aAaB. Then the rope over the top pulley ensures ap=aC and you are there.



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